LEC 2 ORal patholgy chemistry by nhy.pptx

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1. Lecture no 2 Topics covered Predicates and Quantifiers Introduction to Proofs Proofs Methodology

2. Predicates And Quantifiers A statement “ Every computer connected to the university network is functioning properly ” No rules of propositional logic allow us to conclude the truth of the statement “ MATH3 is functioning properly ” Where MATH3 is one of the computers connected to the university network. “enable us to reason with statements that assert that a certain property holds for all objects of a certain type and with statements that assert the existence of an object with a particular property”

3. Predicates Statements involving variables, such as “ x > 3 , ” “ x = y + 3,” “ x + y = z ,” and “ computer x is under attack by an intruder ,” and “ computer x is functioning properly, ” are often found in mathematical assertions, in computer programs, and in system specifications. These statements are neither true nor false when the values of the variables are not specified.

4. “ x is greater than 3 ” has two parts The variable x , is the subject of the statement. The second part the predicate , “ is greater than 3 ” The statement P(x) is also said to be the value of the propositional function P at x . Once a value has been assigned to the variable x , the statement P(x) becomes a proposition and has a truth value. EXAMPLE 1 Let P(x) denote the statement “ x > 3 .” What are the truth values of P( 4 ) and P( 2 ) ? Solution: We obtain the statement P( 4 ) by setting x = 4 in the statement “ x > 3.” Hence P( 4 ) , which is the statement “4 > 3,” is true. However, P( 2 ) , which is the statement “2 > 3,” is false.

5. EXAMPLE 2 Let A(x) denote the statement “ Computer x is under attack by an intruder. ” Suppose that of the computers on campus, only CS2 and MATH1 are currently under attack by intruders . What are truth values of A (CS1), A (CS2), and A (MATH1)? Solution: We obtain the statement A (CS1) by setting x = CS1 in the statement “ Computer x is under attack by an intruder .” Because CS1 is not on the list of computers currently under attack, we conclude that A (CS1) is false . Similarly, because CS2 and MATH1 are on the list of computers under attack, we know that A (CS2) and A (MATH1) are true.

6. EXAMPLE 3 Let Q(x, y) denote the statement “ x = y + 3 .” What are the truth values of the propositions Q( 1 , 2 ) and Q( 3 , ) ? Solution: To obtain Q( 1 , 2 ) , set x = 1 and y = 2 in the statement Q(x, y) . Hence, Q( 1 , 2 ) is the statement “ 1 = 2 + 3 ,” which is false . The statement Q( 3 , ) is the proposition “ 3 = 0 + 3 ,” which is true .

7. Quantifiers Quantification expresses the extent to which a predicate is true over a range of elements . In English, the words all , some , many , none , and few are used in quantifications. The universal quantification of P(x) is the statement “ P(x) for all values of x in the domain .” The notation ∀ xP (x) denotes the universal quantification of P(x) . Here ∀ is called the universal quantifier. We read ∀ xP (x) as “for all xP (x) ” or “for every xP (x) .” An element for which P(x) is false is called a counterexample of ∀ xP (x) .

8. EXAMPLE 8 Let P(x) be the statement “ x + 1 > x .” What is the truth value of the quantification ∀ xP (x) , where the domain consists of all real numbers? Solution: Because P(x) is true for all real numbers x , the quantification ∀ xP (x) is true. Note: Besides “for all” and “for every,” universal quantification can be expressed in many other ways, including “all of,” “for each,” “given any,” “for arbitrary,” “for each,” and “for any.”

9. EXAMPLE 10 Suppose that P(x) is “ > .” To show that the statement ∀ xP (x) is false where the universe of discourse consists of all integers, We give a counterexample. We see that x = 0 is a counterexample because = 0 when x = 0 , so that is not greater than 0 when x = 0. The existential quantification of P(x) is the proposition “ There exists an element x in the domain such that P(x) .” We use the notation ∃ xP (x) for the existential quantification of P(x) . Here ∃ is called the existential quantifier .  

10. EXAMPLE 14 Let P(x) denote the statement “ x > 3 .” What is the truth value of the quantification ∃ xP (x) , where the domain consists of all real numbers? Solution: Because “ x > 3” is sometimes true for instance, when x = 4, the existential quantification of P(x) , which is ∃ xP (x) , is true. Observe that the statement ∃ xP (x) is false if and only if there is no element x in the domain for which P(x) is true. That is, ∃ xP (x) is false if and only if P(x) is false for every element of the domain .

11. EXAMPLE 23 Express the statement “ Every student in this class has studied calculus ” using predicates and quantifiers. Solution: First, we rewrite the statement so that we can clearly identify the appropriate quantifiers to use. Doing so, we obtain: “ For every student in this class, that student has studied calculus .” Next, we introduce a variable x so that our statement becomes “ For every student x in this class, x has studied calculus .”

12. EXAMPLE 24 Express the statements “ Some student in this class has visited Mexico ” and “ Every student in this class has visited either Canada or Mexico ” using predicates and quantifiers. Solution: The statement “ Some student in this class has visited Mexico ” means that “ There is a student in this class with the property that the student has visited Mexico .” We can introduce a variable x , so that our statement becomes “ There is a student x in this class having the property that x has visited Mexico .”

13. Introduction To Proofs Theorem is a statement that can be shown to be true. In mathematical writing, the term theorem is usually reserved for a statement that is considered at least somewhat important. Less important theorems sometimes are called propositions . A proof is a valid argument that establishes the truth of a theorem . The statements used in a proof can include axioms (or postulates ) A less important theorem that is helpful in the proof of other results is called a lemma (plural lemmas or lemmata ).

14. A corollary is a theorem that can be established directly from a theorem that has been proved. A conjecture is a statement that is being proposed to be a true statement, usually on the basis of some partial evidence, a heuristic argument, or the intuition of an expert. When a proof of a conjecture is found, the conjecture becomes a theorem. “ If x > y , where x and y are positive real numbers, then .” really means “ For all positive real numbers x and y , if x > y , then .” Furthermore, when theorems of this type are proved, the first step of the proof usually involves selecting a general element of the domain. Subsequent steps show that this element has the property in question. Finally, universal generalization implies that the theorem holds for all members of the domain.  

15. Methods of Proving Theorems A direct proof of a conditional statement p → q is constructed when the first step is the assumption that p is true; subsequent steps are constructed using rules of inference, with the final step showing that q must also be true. A direct proof shows that a conditional statement p → q is true by showing that if p is true, then q must also be true, so that the combination p true and q false never occurs. Definition: The integer n is even if there exists an integer k such that n = 2 k , and n is odd if there exists an integer k such that n = 2 k + 1. (Note that every integer is either even or odd, and no integer is both even and odd.)

16. EXAMPLE 1 Give a direct proof of the theorem “ If n is an odd integer, then is odd .” Solution: Note that this theorem states ∀ nP ((n) → Q(n)) , where P(n) is “ n is an odd integer ” and Q(n) is “ is odd .” By the definition of an odd integer, it follows that n = 2 k + 1, where k is some integer. We want to show that is also odd. We can square both sides of the equation n = 2 k + 1 to obtain a new equation that expresses . When we do this, we find that . By the definition of an odd integer, we can conclude that is an odd integer (it is one more than twice an integer).Consequently, we have proved that if n is an odd integer, then is an odd integer.  

17. Proof By Contradiction Suppose we want to prove that a statement p is true. Furthermore, suppose that we can find a contradiction q such that ￢ p → q is true. Because q is false, but ￢ p → q is true, we can conclude that ￢ p is false, which means that p is true. How can we find a contradiction q that might help us prove that p is true.

18. EXAMPLE 11 Give a proof by contradiction of the theorem “If 3 n + 2 is odd, then n is odd.” Solution: Let p be “3 n + 2 is odd” and q be “ n is odd.” To construct a proof by contradiction, assume that both p and ￢ q are true. That is, assume that 3 n + 2 is odd and that n is not odd. Because n is not odd, we know that it is even. Because n is even, there is an integer k such that n = 2 k . This implies that 3 n + 2 = 3 ( 2 k) + 2 = 6 k + 2 = 2 ( 3 k + 1 ) . Because 3 n + 2 is 2 t , where t = 3 k + 1, 3 n + 2 is even. Note that the statement “3 n + 2 is even” is equivalent to the statement ￢ p , because an integer is even if and only if it is not odd. Because both p and ￢ p are true, we have a contradiction. This completes the proof by contradiction, proving that if 3 n + 2 is odd, then n is odd.

19. Direct and Indirect Proofs: An Overview Direct Proofs Lead from the premises of a theorem to the conclusion. Begin with premises, followed by deductions, ending in the conclusion. Often, direct proofs may encounter dead ends. Need for Indirect Proofs Used when direct proofs fail. Common for theorems of the form ∀x(P(x) → Q(x)). Indirect proofs do not start with premises and end with the conclusion .

20. Why Use Proof by Contraposition? Useful when direct proofs are difficult. Illustrates logical equivalence through examples. Proof by Contraposition A valuable form of indirect proof. Based on the equivalence: Conditional: p → q Contrapositive: ¬q → ¬p Steps in Proof by Contraposition : Take ¬q as the premise. Use axioms, definitions, and previously proven theorems. Deduce ¬p must follow.