DiscreteMathematicsfor btechAsWeProgress.pptx

Understand concepts of Mathematical Logic, mechanisms of inference rules for propositional and predicate logic and their applications
Understand the concepts of Sets, Relations, Functions and their applications.
Learn the concepts of Algebraic Structures, basics of counting, Principles of inclusion/exclusion and the pigeonhole methodology.
Understand Generating Functions, Recurrence Relations and various ways of solving them.
Understand basic definitions and properties of graphs and their applications in computer science and engineering

1. Course Objectives: Develop ability to Understand concepts of Mathematical Logic, mechanisms of inference rules for propositional and predicate logic and their applications Understand the concepts of Sets, Relations, Functions and their applications. Learn the concepts of Algebraic Structures, basics of counting, Principles of inclusion/exclusion and the pigeonhole methodology. Understand Generating Functions, Recurrence Relations and various ways of solving them. Understand basic definitions and properties of graphs and their applications in computer science and engineering. Course Outcomes: At the end of the course, student would be able to Distinguish between Propositional Logic and Predicate Logic, deriving valid proofs of inference and checking the validity of inferences. Illustrate by examples the basic terminology of sets, relations, functions and algebraic structures along with their associated operations. Demonstrate basics of counting, principles of permutations, combinations, applying inclusion/exclusion principle and the pigeonhole methodology in solving counting problems. Demonstrate Generating functions, write recurrence relations and apply various techniques solving recurrence relations. Transform a problem in computer science and engineering as a graph to solve it efficiently using concepts of graph theory.

2. UNIT-I Mathematical Logic : Statements and notations, Connectives, Well-formed formulas, Truth Tables, tautology, equivalence implication, Normal forms, Quantifiers, universal quantifiers. Predicates: Predicative logic, Free & Bound variables, Rules of inference, Consistency, proof of contradiction, Automatic Theorem Proving. UNIT-II Relations: Properties of Binary Relations, equivalence, transitive closure, compatibility and partial ordering relations, Lattices, Hasse diagram. Functions: Inverse Function Composition of functions, recursive Functions, Lattice and its Properties UNIT-III Elementary Combinatorics: Basis of counting, Combinations & Permutations, with repetitions, Constrained repetitions, Binomial Coefficients, Binomial Multinomial theorems, the principles of Inclusion – Exclusion. Pigeon hole principles and its application UNIT-IV Recurrence Relation : Generating Functions, Function of Sequences Calculating Coefficient of generating function, Recurrence relations, solving recurrence relation by substitution and Generating funds. Characteristics solution of inhomogeneous Recurrence Relation. UNIT-V Graph Theory: Basic Concepts, Representation of Graph, Isomorphism, Sub graphs, Spanning Trees, Planar Graphs, Multi graphs, Euler circuits, Hamiltonian graphs, Chromatic Numbers. Graph Theory and Applications.

3. Lesson plan

4. Suggested Readings: Discrete Mathematical Structures with Applications to Computer Science, J.P. Tremblay, R. Manohar, McGraw Hill education (India) Private Limited. (UNITS - I , II ) Discrete Mathematics for Computer Scientists & Mathematicians, Joe L. Mott, Abraham Kandel, Theodore P. Baker, Pearson , 2nd ed. (Units - III, IV, V ) Elements of Discrete Mathematics- A Computer Oriented Approach- C L Liu, D P Mohapatra. Third Edition, Tata McGrawHill . Discrete Mathematics for Computer Scientists & Mathematicians, J.L. Mott, A.Kandel , T.P. Baker, PHI. Discrete Mathematics and its Applications, Kenneth H. Rosen, Fifth Edition.TMH . Discrete Mathematical Structures Theory and Application-Malik & Sen, Cengage.

5. What is Discrete Mathematics: Discrete mathematics is the branch of mathematics dealing with objects that can assume only distinct, separated values. Unlike continuous mathematics, which studies concepts such as calculus and real numbers that can vary smoothly, discrete mathematics focuses on discrete structures such as integers, graphs, and logic. It involves topics like set theory, combinatorics, graph theory, logic, and algorithms, all of which have finite or countable structures.

6. Why We Study Discrete Mathematics: Foundation of Computer Science: Discrete mathematics is essential in computer science because it provides the mathematical foundation for understanding algorithms, data structures, cryptography, and the design of computer networks. Many computational problems, such as sorting and searching, are solved using discrete mathematical techniques. Algorithm Development: Discrete math allows us to understand the logic and efficiency of algorithms, which are step-by-step procedures for solving problems. It helps in analyzing the time complexity and optimization of algorithms. Modeling Real-World Problems: It is used to model problems in various fields like biology, economics, social networks, and optimization. Problems involving networks, scheduling, or resource allocation often rely on discrete mathematics. Logical Thinking and Problem Solving: Discrete mathematics enhances logical reasoning and problem-solving skills. It helps in formalizing arguments, constructing proofs, and thinking critically, which are essential in both academic research and industry. Applications in Cryptography: Discrete mathematics, particularly number theory and combinatorics, plays a crucial role in cryptography, which is vital for secure communication in fields like finance, defense , and digital technologies. In summary, we study discrete mathematics because it is fundamental to many modern technologies, particularly in computer science, and it fosters critical thinking, precision, and problem-solving skills crucial for tackling complex problems.

7. UNIT-1   Mathematical Logic: Logic It is the study of the principles and methods that distinguishes between valid and invalid argument.   Proposition Logic   Proposition or Statement A proposition or a statement can defined has a declarative sentence to which we can assign one and only one of the truth values either true (or) false but not a both is called a proposition. The true or false of a proposition is called truth value of a proposition   These two values true and false are denoted by the symbols T and F respectively. Sometimes these are also denoted by the symbols 1 and 0 respect. Ex: 1) India capital is new Delhi  True 2*3=5 false 5 is a prime number true These are propositions (or statements) because they are either true or false. Next consider the following sentences: How beautiful are you? Wish you a happy new year x + y = z Take one book. These are not propositions as they are not declarative in nature, that is, they do not declare a definite truth value T or F .  

8. Types of Propositions 1)Atomic proposition 2)Compound Proposition 1)Atomic proposition           Notations Statements are symbolically represented as A,B,C,..P,Q,R,S… Those are called propositional variables or notations. Examples: P=―Delhi is the capital of India‖. Q=―17 is divisible by 3. A Proposition which cannot be divided further is called an atomic proposition . Examples: 1)India’s capital is New Delhi 2)2*3=5   2) Compound Proposition • Two or more atomic propositions can be combined to form a compound proposition with help of Connectives. Compound Proposition also called as Propositional function. Examples of Compound statements ―3+2=5‖ and ―Delhi is a capital of India‖. ―the grass is green‖ or ―it is hot today‖. Discrete mathematics is not difficult to me. Here and, or ,not are called connectives.  

9. Connectives The words or phrases or symbols which are used to make a compound proposition by two or more atomic propositions are called logical connectives or simply connectives. There are five basic connectives called negation, conjunction, disjunction, conditional and biconditional.   Negation (~) or(¬) The negation of a statement is generally formed by writing the word ‗ not ‗ at a proper place in the statement (proposition) or by prefixing the statement with the phrase (~).It is not the case that ‗ . If p denotes a statement then the negation of p is written as p and read as not p ‗ . If the truth value of p is T then the truth value of p is F . Also if the truth value of p is F then the truth value of p is T . Connectivity symbol   Word Negation ~   Not Disjunction ∨   OR Conjunction ∧   AND Conditional →   IF AND THEN Biconditional   ↔   If and only if   Ex:~P Truth table for Negation P ~P F T

10. Disjunction (OR) If P and Q are any two propositions then P or Q symbolically written as P V Q . P V Q is a proposition whose truth values is false only when both P and Q are false other wise True. Truth table for Disjunction P Q P V Q F F F F T T T F T T T T P: I shall go to the game. Q: I shall watch the game on television. P V Q: I shall go to the game OR I shall watch the game on television.

11. Conjunction (AND(^)) IF P and Q are any two propositions then P and Q Symbolically written as P ^ Q. P ^ Q is a proposition whose truth value is true only when both are P and Q are true. Whole truth value is false when either P or Q are false and both are false. P Q P^ Q F F F F T F T F F T T T Truth table for Conjunction P : It is raining today. Q:There are 10 chairs in the room. P ^ Q: It is raining today. ANDThere are 10 chairs in the room. ^

12. Conditi onal (or) Implication(→) If P and Q are any two statements (or propositions) then the statement P → Q which is read as, If P , then Q ‗ is called a conditional statement (or proposition) or implication and the connective is the conditional connective. Truth table for conditional P Q   P→ Q   F F T F T T T F F T   T   T   In this conditional statement, P is called the hypothesis or premise or antecedent and Q is called the consequence or conclusion.   Biconditional (If and only if) If P and Q are any two propostions then P if and only if Q Written as P ↔Q. It‘s truth value is true only when both P & Q have same truth values. P q p↔q T T T T F F F T F F F T

13. TAUTOLOGY AND CONTRADICTION Tautology: A proposition is said to be a tautology if its truth value is T for any assignment of truth values to its components. Example: The proposition P V ~P is a tautology Contradiction A proposition is said to be a contradiction if its truth value is F for any assignment of truth values to its components. Example: The proposition P ^ ¬ P is a contradiction Contingency : A statement formula which is neither a tautology nor a contradiction is knownas a contingency. Example: P->Q   Tautology: A statement formula which is true regardless of the truth values of the statements which replace the variables in it is called a universally valid formula or a logical truth or a tautology. How to prove given compound proposition is tautology 1)By Constructing truth table 2)By using substitution method

14. 1)Constructing truth table Show that following function is tautology a)(PVQ) V ~P Implication: If P and Q are any two propositions then P->Q or If P then Q P->Q is proposition whose truth value is false only when P is true and Q is false. P->Q Here P is antecedent and Q is Consequent. P   Q P→ Q   F F T F T T T F F T T T from the implication statement we can write another three statements which are converse, inverse and contrapositive Converse, Inverse and Contrapositive If P→Q is a conditional statement, then (1). Q→P is called its converse (3). ¬Q→ ¬P is called its contrapositive. (2). ¬P→ ¬Q is called its inverse

15. Example: • P: Today is Sunday • Q: It is a holiday   Converse Statement: If it is a holiday, then today is Sunday. Inverse Statement: If today is not Sunday, then it is not a holiday.   Contrapositive Statement- If it is not a holiday, then today is not Sunday.   Here Implication and Contrapositive are equal. Converse and inverse are opposite propositions.

16. Modus Ponens (Affirming the Antecedent) Definition: Modus Ponens is a rule of inference in propositional logic that follows this structure: If P→QP \ rightarrow QP→Q (If PPP, then QQQ). PPP (Premise: PPP is true). Therefore, QQQ (Conclusion: QQQ is true). Example: If it rains, then the ground will be wet. ( P→QP \ rightarrow QP→Q) It is raining. ( PPP) Therefore, the ground is wet. ( QQQ) Explanation: In Modus Ponens, if we know that PPP implies QQQ and that PPP is true, then we can conclude that QQQ must also be true.

17. 2. Modus Tollens (Denying the Consequent) Definition: Modus Tollens is another valid form of inference but works by denying the consequent: If P→QP \ rightarrow QP→Q (If PPP, then QQQ). ¬Q\neg Q¬Q (Premise: QQQ is false). Therefore, ¬P\neg P¬P (Conclusion: PPP is false). Example: If it rains, then the ground will be wet. ( P→QP \ rightarrow QP→Q) The ground is not wet. ( ¬Q\neg Q¬Q) Therefore, it did not rain. ( ¬P\neg P¬P) Explanation: In Modus Tollens, if we know that PPP implies QQQ, and we observe that QQQ is false, we can conclude that PPP must also be false. Key Differences: Modus Ponens affirms the antecedent (PPP) and concludes that the consequent (QQQ) must also be true. Modus Tollens denies the consequent (QQQ) and concludes that the antecedent (PPP) must be false. Both are fundamental rules of logical inference and are widely used in formal logic, mathematics, and philosophical reasoning. 4o

18. Syllogism is a form of deductive reasoning in logic, where a conclusion is drawn from two given or assumed propositions (premises). The typical form of a syllogism consists of three parts: a major premise , a minor premise , and a conclusion . The conclusion follows logically from the premises. Structure of a Syllogism: A syllogism has the following structure: Major Premise : A general statement or universal truth. Minor Premise : A specific statement that falls under the general rule of the major premise. Conclusion : The statement that logically follows from the two premises. Example of a Syllogism: Major Premise : All humans are mortal. Minor Premise : Socrates is a human. Conclusion : Therefore, Socrates is mortal. Here, the major premise establishes a general truth ("All humans are mortal"), the minor premise identifies a specific case ("Socrates is a human"), and the conclusion logically deduces that Socrates must also be mortal.

19. Types of Syllogism: Categorical Syllogism : In this type, both premises and the conclusion are categorical propositions (statements that assert something about all or some members of a category). Example: Major Premise: All dogs are animals. Minor Premise: All poodles are dogs. Conclusion: Therefore, all poodles are animals. Conditional (Hypothetical) Syllogism : In this form, the major premise is a conditional statement (an "if-then" statement). Example: Major Premise: If it rains, the ground will be wet. Minor Premise: It is raining. Conclusion: Therefore, the ground is wet. Disjunctive Syllogism : This form involves a major premise that presents an "either-or" situation (a disjunction). Example: Major Premise: Either the train is late or it has already arrived. Minor Premise: The train is not late. Conclusion: Therefore, it has already arrived.

20. Rules of a Valid Syllogism: Three Terms Rule : A valid syllogism must contain exactly three terms: the major term, minor term, and middle term. Middle Term Distribution : The middle term must be distributed (cover all members of the category) in at least one of the premises. Negation : If either premise is negative, the conclusion must be negative. If both premises are affirmative, the conclusion must also be affirmative. Importance of Syllogism: Foundation of Logic : Syllogism is one of the most basic forms of logical argument, forming the foundation for many logical and philosophical systems. Critical Thinking : Understanding syllogism improves deductive reasoning and critical thinking skills, helping to identify valid and invalid arguments. Problem-Solving : It is widely used in fields like mathematics, law, computer science, and philosophy for systematic problem-solving and argument analysis. In summary, syllogism is an important tool in formal logic, helping to derive conclusions from given premises in a structured and systematic way.

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22. A binary relation on a set is a collection of ordered pairs of elements from the set. More formally, if AAA is a set, then a binary relation RRR on AAA is a subset of A×AA \times AA×A (the Cartesian product of AAA with itself). The properties of a binary relation can vary depending on its characteristics. Here are some common properties of binary relations: 1. Reflexive A relation RRR on a set AAA is reflexive if every element of AAA is related to itself. Formally: ∀ a∈A ,( a,a )∈R.\ forall a \in A, (a, a) \in R.∀ a∈A ,( a,a )∈R. Example : The relation R={(1,1),(2,2)}R = \{(1, 1), (2, 2)\}R={(1,1),(2,2)} on the set A={1,2}A = \{1, 2\}A={1,2} is reflexive because every element of AAA is related to itself. 2. Symmetric A relation RRR is symmetric if for all elements aaa and bbb in AAA, whenever aaa is related to bbb , bbb is also related to aaa . Formally: ∀ a,b∈A , ( a,b )∈R  ⟹  ( b,a )∈R.\ forall a, b \in A, \ (a, b) \in R \implies (b, a) \in R.∀ a,b∈A , ( a,b )∈R⟹( b,a )∈R. Example : If the relation is "is a sibling of," then it is symmetric because if aaa is a sibling of bbb , then bbb is also a sibling of aaa . 3. Anti-Symmetric A relation RRR is anti-symmetric if for all elements aaa and bbb in AAA, whenever aaa is related to bbb and bbb is related to aaa , then aaa must be equal to bbb . Formally: ∀ a,b∈A , ( a,b )∈R and ( b,a )∈R  ⟹  a=b.\ forall a, b \in A, \ (a, b) \in R \ \text{and} \ (b, a) \in R \implies a = b.∀ a,b∈A , ( a,b )∈R and ( b,a )∈ R⟹a =b. Example : The "less than or equal to" (≤\ leq ≤) relation is anti-symmetric because if a≤ba \ leq ba≤b and b≤ab \ leq ab≤a , then a= ba = ba =b.

23. 4. Transitive A relation RRR is transitive if for all elements aaa , bbb , and ccc in AAA, whenever aaa is related to bbb and bbb is related to ccc, then aaa must also be related to ccc. Formally: ∀ a,b,c∈A , ( a,b )∈R and ( b,c )∈R  ⟹  ( a,c )∈R.\ forall a, b, c \in A, \ (a, b) \in R \ \text{and} \ (b, c) \in R \implies (a, c) \in R.∀ a,b,c∈A , ( a,b )∈R and ( b,c )∈R⟹( a,c )∈R. Example : The "is an ancestor of" relation is transitive because if aaa is an ancestor of bbb and bbb is an ancestor of ccc, then aaa is an ancestor of ccc. 5. Irreflexive A relation RRR is irreflexive if no element of AAA is related to itself. Formally: ∀ a∈A , ( a,a )∉R.\ forall a \in A, \ (a, a) \ notin R.∀ a∈A , ( a,a )∈/R. Example : The "less than" (<<<) relation is irreflexive because no number is less than itself. 6. Asymmetric A relation RRR is asymmetric if for all elements aaa and bbb in AAA, whenever aaa is related to bbb , bbb is never related to aaa . Formally: ∀ a,b∈A , ( a,b )∈R  ⟹  ( b,a )∉R.\ forall a, b \in A, \ (a, b) \in R \implies (b, a) \ notin R.∀ a,b∈A , ( a,b )∈R⟹( b,a )∈/R. Example : The "less than" (<<<) relation is asymmetric because if a< ba < ba <b, then it cannot be true that b<ab < ab<a. 7. Total (or Connex) A relation RRR on a set AAA is total if every pair of elements in AAA is comparable. That is, for any aaa and bbb in AAA, either ( a,b )∈R(a, b) \in R( a,b )∈R or ( b,a )∈R(b, a) \in R( b,a )∈R. Formally: ∀ a,b∈A ,  a≠b   ⟹  ( a,b )∈R or ( b,a )∈R.\ forall a, b \in A, \ a \ neq b \implies (a, b) \in R \ \text{or} \ (b, a) \in R.∀ a,b∈A , a=b⟹( a,b )∈R or ( b,a )∈R. Example : The "less than or equal to" (≤\ leq ≤) relation on the set of real numbers is total, because for any two real numbers aaa and bbb , either a≤ba \ leq ba≤b or b≤ab \ leq ab≤a .

24. 8. Equivalence Relation A relation RRR is an equivalence relation if it is reflexive, symmetric, and transitive. An equivalence relation partitions a set into equivalence classes. Example : The relation "is congruent to" in geometry or "has the same remainder when divided by a number" in modular arithmetic are both examples of equivalence relations. 9. Partial Order A relation RRR is a partial order if it is reflexive, anti-symmetric, and transitive. A set with a partial order is called a partially ordered set ( poset ). Example : The "subset" relation (⊆\ subseteq ⊆) between sets is a partial order because it satisfies these three properties. 10. Total Order A relation is a total order if it is a partial order and it is total ( connex ). In other words, it satisfies reflexivity, anti-symmetry, transitivity, and comparability. Example : The "less than or equal to" (≤\ leq ≤) relation on real numbers is a total order. These are the key properties of binary relations that allow us to classify and analyze relationships between elements of sets.

25. Composition of Binary Relations The composition of binary relations is a fundamental concept in mathematics, particularly in set theory and relation theory. It involves combining two relations to form a new relation that directly relates elements of the first set to elements of the last set via an intermediate set. Definition: Given two binary relations: R⊆A×BR \ subseteq A \times BR⊆A×B, a relation from set AAA to set BBB, S⊆B×CS \ subseteq B \times CS⊆B×C, a relation from set BBB to set CCC, the composition of RRR and SSS, denoted by R∘SR \ circ SR∘S, is a new relation from AAA to CCC, defined as: R∘S={( a,c )∈A×C∣∃ b∈B , ( a,b )∈R and ( b,c )∈S}.R \ circ S = \{(a, c) \in A \times C \mid \exists b \in B, \ (a, b) \in R \text{ and } (b, c) \in S \}.R∘S={( a,c )∈A×C∣∃ b∈B , ( a,b )∈R and ( b,c )∈S}.In simpler terms, the pair ( a,c )(a, c)( a,c ) belongs to R∘SR \ circ SR∘S if there exists an element b∈Bb \in Bb∈B such that aaa is related to bbb by RRR and bbb is related to ccc by SSS.

26. How It Works: To understand the composition of relations, consider the following step-by-step process: Starting with Relation RRR : RRR connects an element a∈Aa \in Aa∈A to some b∈Bb \in Bb∈B . Following with Relation SSS : SSS connects the same element b∈Bb \in Bb∈B to some c∈Cc \in Cc∈C . Composing RRR and SSS : The composition R∘SR \ circ SR∘S directly connects a∈Aa \in Aa∈A to c∈Cc \in Cc∈C if aaa is related to ccc through bbb in the intermediate set BBB. Example: Let A={1,2}A = \{1, 2\}A={1,2}, B={3,4}B = \{3, 4\}B={3,4}, and C={5,6}C = \{5, 6\}C={5,6}, and consider the following relations: R={(1,3),(2,4)}⊆A×BR = \{(1, 3), (2, 4)\} \ subseteq A \times BR={(1,3),(2,4)}⊆A×B, S={(3,5),(4,6)}⊆B×CS = \{(3, 5), (4, 6)\} \ subseteq B \times CS={(3,5),(4,6)}⊆B×C. The composition R∘SR \ circ SR∘S is computed as follows: (1,3)∈R(1, 3) \in R(1,3)∈R and (3,5)∈S(3, 5) \in S(3,5)∈S, so (1,5)∈R∘S(1, 5) \in R \ circ S(1,5)∈R∘S. (2,4)∈R(2, 4) \in R(2,4)∈R and (4,6)∈S(4, 6) \in S(4,6)∈S, so (2,6)∈R∘S(2, 6) \in R \ circ S(2,6)∈R∘S. Thus, R∘S={(1,5),(2,6)}R \ circ S = \{(1, 5), (2, 6)\}R∘S={(1,5),(2,6)}.

27. Properties of Composition: Associativity : Composition of relations is associative. For any three relations RRR, SSS, and TTT, the following holds: (R∘S)∘T=R∘(S∘T).(R \ circ S) \ circ T = R \ circ (S \ circ T).(R∘S)∘T=R∘(S∘T).This means that when composing multiple relations, the order of operations does not matter. Identity Relation : The identity relation IAI_AIA​ on set AAA is defined as IA={( a,a )∣ a∈A }I_A = \{(a, a) \mid a \in A\}IA​={( a,a )∣ a∈A }. The composition of any relation RRR with the identity relation IAI_AIA​ on the corresponding set leaves RRR unchanged: R∘IA=R=IB∘R,R \ circ I_A = R = I_B \ circ R,R∘IA​=R=IB​∘ R,where IBI_BIB​ is the identity relation on set BBB. Domain and Range : The domain of R∘SR \ circ SR∘S is the set of all elements a∈Aa \in Aa∈A for which there exists an element c∈Cc \in Cc∈C such that ( a,c )∈R∘S(a, c) \in R \ circ S( a,c )∈R∘S. The range of R∘SR \ circ SR∘S is the set of all elements c∈Cc \in Cc∈C for which there exists an element a∈Aa \in Aa∈A such that ( a,c )∈R∘S(a, c) \in R \ circ S( a,c )∈R∘S. Non-Commutativity : In general, the composition of relations is not commutative, meaning R∘S≠S∘RR \ circ S \ neq S \ circ RR∘S=S∘R in most cases.

28. Applications: Databases : In relational databases, the composition of relations can be used to join tables based on common fields. Graph Theory : Composition is used to determine paths between nodes in a graph. Formal Language Theory : In automata theory, the composition of relations can model state transitions. Conclusion: The composition of binary relations is a powerful and versatile operation that plays a crucial role in various fields, including mathematics, computer science, and logic. It allows us to combine simple relations to express more complex relationships and is fundamental in understanding the structure and behavior of relational systems.

29. Transitive Closure of Relations In mathematics, particularly in the study of relations, the transitive closure of a binary relation is an important concept. It extends a given relation to include all possible transitive relations. Definition: Given a binary relation RRR on a set AAA, the transitive closure of RRR, denoted as R+, is the smallest transitive relation on AAA that contains RRR. In other words, R+ adds to RRR all pairs ( a,c )(a, c)( a,c ) such that there is a sequence of intermediate elements linking aaa to ccc through RRR. For a relation RRR to be transitive, whenever ( a,b )∈R(a, b) \in R( a,b )∈R and ( b,c )∈R(b, c) \in R( b,c )∈R, it must be that ( a,c )∈R(a, c) \in R( a,c )∈R. What does it mean? If RRR is a binary relation on set AAA, then the transitive closure adds all those "missing" connections that ensure the relation is transitive. A relation RRR is transitive if, whenever an element is related to a second element, and the second element is related to a third, the first element is directly related to the third. Example: If aaa is related to bbb , and bbb is related to ccc, the transitive closure ensures that aaa is also directly related to ccc.

30. Formal Expression: Let RRR be a binary relation on set AAA, then the transitive closure R+R^+R+ is: R+=R∪R2∪R3∪…R^+ = R \cup R^2 \cup R^3 \cup \ dotsR +=R∪R2∪R3∪…Where: R2R^2R2 is the composition of RRR with itself. R3R^3R3 is the composition of R2R^2R2 with RRR, and so on. In simpler terms, the transitive closure of RRR adds to RRR all possible relations that can be formed by following multiple steps along RRR. Example: Let’s consider an example with a set A={1,2,3}A = \{1, 2, 3\}A={1,2,3} and a relation R={(1,2),(2,3)}R = \{(1, 2), (2, 3)\}R={(1,2),(2,3)}. In this case, RRR is not transitive because (1,2)∈R(1, 2) \in R(1,2)∈R and (2,3)∈R(2, 3) \in R(2,3)∈R, but (1,3)∉R(1, 3) \ notin R(1,3)∈/R. The transitive closure of RRR would be R+={(1,2),(2,3),(1,3)}R^+ = \{(1, 2), (2, 3), (1, 3)\}R+={(1,2),(2,3),(1,3)}, because now it includes the pair (1,3)(1, 3)(1,3), making it transitive. Steps to Find the Transitive Closure: Start with the given relation RRR. Check for transitive property : For each pair ( a,b )∈R(a, b) \in R( a,b )∈R and ( b,c )∈R(b, c) \in R( b,c )∈R, check if ( a,c )∈R(a, c) \in R( a,c )∈R. If not, add it to the relation. Repeat : Continue this process until no more new pairs can be added.

31. Warshall’s Algorithm (Efficient way to find Transitive Closure): An efficient algorithm to compute the transitive closure of a relation represented by an adjacency matrix is Warshall’s Algorithm . The steps are: Start with the adjacency matrix of the given relation. For each intermediate node kkk , update the matrix to include paths passing through kkk . The resulting matrix will represent the transitive closure of the relation. Real-life Example of Transitive Closure: Consider a social network where people are connected by the relation "is a friend of." If person AAA is a friend of person BBB, and person BBB is a friend of person CCC, then the transitive closure of this relation would add the fact that person AAA is indirectly a friend of person CCC (through BBB). Thus, the transitive closure helps in identifying chains of relationships, indicating who can be connected through intermediaries. This can be useful in understanding reachability in networks, communication systems, and transportation routes. Applications of Transitive Closure: Database Queries : Finding transitive relations like "If person AAA manages person BBB, and person BBB manages person CCC, then person AAA manages person CCC". Graph Theory : In a directed graph, the transitive closure tells us which vertices are reachable from which others. Networking : Transitive closure is used to analyze reachability in communication networks. Formal Logic and AI : In reasoning systems, transitive closure helps deduce new facts from known relationships. In summary, the transitive closure of a relation adds all the missing links needed to make the relation transitive, ensuring that indirect relationships are included in the original relation.

32. 1. Compatibility Relations: A compatibility relation is a relation that describes when two elements of a set can "coexist" or are "compatible" in a certain sense. The definition of compatibility varies depending on the specific application, but it generally means that two elements can interact or be combined under certain operations without violating defined rules. In many mathematical structures, compatibility refers to elements maintaining certain properties (e.g., symmetry, associativity) when combined. For instance, in algebra, two operations might be said to be compatible if applying one after the other results in a well-defined result according to the structure’s rules. Properties of Compatibility Relations (depending on context): Symmetry : If aaa is compatible with bbb , then bbb is compatible with aaa . Reflexivity : Every element is compatible with itself. Transitivity : If aaa is compatible with bbb , and bbb is compatible with ccc, then aaa is compatible with ccc. Example : In a vector space, two vectors might be said to be compatible if they are linearly independent, allowing them to form a basis for the space.

33. 2. Partial Ordering Relations: A partial order is a binary relation ≤\ leq ≤ on a set AAA that satisfies the following properties: Reflexivity : Every element is related to itself. For all a∈Aa \in Aa∈A , a≤aa \ leq aa≤a . Antisymmetry : If two elements are mutually related, they must be the same element. If a≤ba \ leq ba≤b and b≤ab \ leq ab≤a , then a= ba = ba =b. Transitivity : If one element is related to a second, and the second is related to a third, then the first must be related to the third. If a≤ba \ leq ba≤b and b≤cb \ leq cb≤c , then a≤ca \ leq ca≤c . Partial Order Example: Let’s take an example with the set A={1,2,3,4}A = \{1, 2, 3, 4\}A={1,2,3,4} and define the binary relation ≤\ leq ≤ as the "divides" relation, where a≤ba \ leq ba≤b if aaa divides bbb . For example: 1≤21 \ leq 21≤2 because 1 divides 2, 1≤31 \ leq 31≤3 because 1 divides 3, 2≤42 \ leq 42≤4 because 2 divides 4, 1≤41 \ leq 41≤4 because 1 divides 4. This relation satisfies the three properties of a partial order: Reflexive : Every element divides itself (e.g., 2≤22 \ leq 22≤2). Antisymmetric : If a≤ba \ leq ba≤b and b≤ab \ leq ab≤a , then a= ba = ba =b. Transitive : If a≤ba \ leq ba≤b and b≤cb \ leq cb≤c , then a≤ca \ leq ca≤c .

34. Hasse Diagram for Partial Orders: A Hasse diagram is a graphical representation of a partial order. It visualizes elements of a set and shows the ordering relation between them. The diagram omits reflexive loops and minimizes arrows by only showing the most direct relationships. For example, if a≤ba \ leq ba≤b and b≤cb \ leq cb≤c , the Hasse diagram would show an arrow from aaa to bbb , and from bbb to ccc, but not directly from aaa to ccc (the transitivity is implicit). Real-life Example of Partial Ordering: Consider a set of tasks that must be completed in a specific order: Task A must be done before Task B, Task B must be done before Task C, Task A must also be done before Task C. This defines a partial order on the set of tasks, where the ordering reflects the prerequisite structure of tasks. The relation ≤\ leq ≤ in this case describes which tasks must precede others. Summary: Compatibility Relations describe when two elements can coexist or interact without conflict. Partial Ordering Relations provide a structured way to compare elements in a set, reflecting hierarchical or dependency relationships, with three key properties: reflexivity, antisymmetry , and transitivity. Both concepts are widely used in mathematics, computer science (for tasks scheduling, process dependencies), and many real-life systems that require structured organization and interaction.