Discrete mathematics [LOGICAL CONNECTIVES]

CONNECTIVES-CONNJUCTION,DISJUNCTION,IMPLICATION,CONTRADICTIONS,TATULOGY,CONTIGENCY...........

1. TOPIC: LOGICAL CONNECTIVES NAME:KALINGO AUROBINDO NAYAK REGD NO:20010112022 BRANCH:CSE CAMPUS:PARALAKHMUNDI

2. Prepositional Logic – Definition A proposition is a collection of declarative statements that has either a truth value "true” or a truth value "false". A propositional consists of propositional variables and connectives. We denote the propositional variables by capital letters (A, B, etc). The connectives connect the propositional variables. Some examples of Propositions are given below − "Man is Mortal", it returns truth value “TRUE” "12 + 9 = 3 – 2", it returns truth value “FALSE” The following is not a Proposition − "A is less than 2". It is because unless we give a specific value of A, we cannot say whether the statement is true or false.

3. Connectives

4. I. Truth Table of Logical Negation Negation ( ¬)  − The negation of a proposition A (written as ¬A) is false when A is true and is true when A is false. LET:- A: the number 2 is greater than 7 ¬A : the number 2 is not greater than 7. A ¬ A True False False True

5. Conjunction(AND) AND ( ∧)  − The AND operation of two propositions A and B (written as A∧B) is true if both the propositional variable A and B is true. A B A ∧ B True True True True False False False True False False False False Example 1 : Let r: 5 be a rational number and s: 15 be a prime number. Is it a conjunction? Solution : Given that r: 5 is a rational number. This proposition is true. s: 15 is a prime number. This proposition is false as 15 is a  composite number . Therefore, as per the truth table, r and s is a false statement. So, r ∧ s = F

6. Disjunction(OR) OR ( ∨)  − The OR operation of two propositions A and B (written as A∨B) is true if at least any of the propositional variable A or B is true. Let P: bus left early and Q: my watch going slow P ∨Q : bus  left early or my watch going slow A B A ∨ B True True True True False True False True True False False False

7. Implication / if-then ( →)   An implication A→B is the proposition “if A, then B”. It is false if A is true and B is false. The rest cases are true. The truth table is as follows − Let P: Aeroplane reaches in time And q: I can attend the meeting .then, P→Q= if Aeroplane reaches in time, then  I can Attend the meeting A B A → B True True True True False False False True True False False True

8. bi-conditional connective If and only if ( ⇔)  − A⇔B is bi-conditional logical connective which is true when p and q are same, i.e. both are false or both are true. The truth table is as follows − Let P: Δ abc is an isosceles triangle And Q: two sides of a triangle are equal Then P  ⇔  Q : Δ abc is an isosceles triangle If and only if  two sides of a triangle are equal A B A ⇔ B True True True True False False False True False False False True

9. Tautologies A Tautology is a formula which is always true for every value of its propositional variables. Example  − Prove [(A→B)∧A]→Bis a tautology The truth table is as follows − As we can see every value of  [( A → B )∧ A ]→ B  is "True", it is a tautology. A B A → B (A → B) ∧ A [( A → B ) ∧ A] → B True True True True True True False False False True False True True False True False False True False True

10. Contradictions A Contradiction is a formula which is always false for every value of its propositional variables. Example  − Prove (A∨B)∧[(¬A)∧(¬B)] is a contradiction A B A ∨ B ¬ A ¬ B (¬ A) ∧ ( ¬ B) (A ∨ B) ∧ [( ¬ A) ∧ (¬ B)] True True True False False False False True False True False True False False False True True True False False False False False False True True True False The truth table is as follows − As we can see every value of  ( A ∨ B )∧[(¬ A )∧(¬ B )]  is “False”, it is a contradiction.

11. Contingency A Contingency is a formula which has both some true and some false values for every value of its propositional variables. Example  − Prove (A∨B)∧(¬A)a contingency As we can see every value of (A∨B)∧(¬A) has both “True” and “False”, it is a contingency A B A ∨ B ¬ A (A ∨ B) ∧ (¬ A) True True True False False True False True False False False True True True True False False False True False

12. Inverse, Converse, and Contra-positive

13. Converse

14. Contra-positive