1. LOGICAL EQUIVALENCES FORHAD AHMED KHAN DISCRETE MATHEMATICS

2. LOGICAL EQUIVALENCES Definition: The compound propositions p and q are said to be logically equivalent if p ↔ q is a Tautology. Compound propositions that have the same truth values in all possible cases are called logically equivalent. Logical Equivalence is denoted by ≡ or  Example p ^T ≡ p  p ^T ≡ p pT p ^T TTT FTF p ^T is equivalent to p. In this example, p ^T is equivalent to p. Lets try to understand with truth table. Let p be the proposition. There are two possible truth values for this proposition, True or False. Here, T means True. So all of it’s cases will be True. Now for p ^T, all possible truth values depends on p. So, p is equivalent to p ^T.

3. Most Common and Famous Logical Equivalences LOGICAL EQUIVALENCES ( DISCRETE MATHEMATICS ) 3

4. 1. Identity Laws a) p ^ T ≡ p pT p ^T TTT FTF b) p v F ≡ p pF p v T TFT FFF LOGICAL EQUIVALENCES ( DISCRETE MATHEMATICS ) 4

5. 5 2. Domination Laws a) p v T ≡ T pT p v T TTT FTT b) p ^ F ≡ F pF p ^ T TFF FFF LOGICAL EQUIVALENCES ( DISCRETE MATHEMATICS )

6. 6 3. Idempotent Laws a) p v p ≡ p pp p v p TTT FFF b) p ^ p ≡ p pp p ^ p TTT FFF LOGICAL EQUIVALENCES ( DISCRETE MATHEMATICS )

7. 7 4. Double Negation Law a) ┐(┐p) ≡ p p┐p┐(┐p) TFT FTF LOGICAL EQUIVALENCES ( DISCRETE MATHEMATICS )

8. 8 5. Commutative Laws - a a) p v q ≡ q v p pq p v q q v p TTTT TFTT FTTT FFFF LOGICAL EQUIVALENCES ( DISCRETE MATHEMATICS )

9. 9 5. Commutative Laws - b b) p ^ q ≡ q ^ p pq p ^ q q ^ p TTTT TFFF FTFF FFFF LOGICAL EQUIVALENCES ( DISCRETE MATHEMATICS )

10. 10 6. Associative Laws - a a) (p v q) v r ≡ p v ( q v r) a) (p v q) v r ≡ p v ( q v r) pqr p v q q v p (p v q) v r p v ( q v r) TTTTTTT TTFTTTT TFTTTTT TFFTTTT FTTTTTT FTFTTTT FFTFFTT FFFFFFF LOGICAL EQUIVALENCES ( DISCRETE MATHEMATICS )

11. 11 6. Associative Laws - b b) (p ^ q) ^ r ≡ p ^ ( q ^ r) b) (p ^ q) ^ r ≡ p ^ ( q ^ r) pqr p ^ q q ^ p (p ^ q) ^ r p ^ ( q ^ r) TTTTTTT TTFTTFF TFTFFFF TFFFFFF FTTFFFF FTFFFFF FFTFFFF FFFFFFF LOGICAL EQUIVALENCES ( DISCRETE MATHEMATICS )

12. 12 7. Distributive Laws - a a) p v ( q ^ r) ≡ (p v q) ^ (p v r) a) p v ( q ^ r) ≡ (p v q) ^ (p v r) pqr p v q p v r q ^ r p v ( q ^ r) (p v q) ^ (p v r) TTTTTTTT TTFTTFTT TFTTTFTT TFFTTFTT FTTTTTTT FTFTFFFF FFTFTFFF FFFFFFFF LOGICAL EQUIVALENCES ( DISCRETE MATHEMATICS )

13. 13 7. Distributive Laws - b b) p ^ ( q v r) ≡ (p ^ q) v (p ^ r) b) p ^ ( q v r) ≡ (p ^ q) v (p ^ r) pqr p ^ q p ^ r q v r p ^ ( q v r) (p ^ q) v (p ^ r) TTTTTTTT TTFTFTTT TFTFTTTT TFFFFFFF FTTFFTFF FTFFFTFF FFTFFTFF FFFFFFFF LOGICAL EQUIVALENCES ( DISCRETE MATHEMATICS )

14. 14 8. De Morgan’s Laws - a a) ┐(p ^ q) ≡ ┐p v ┐q a) ┐(p ^ q) ≡ ┐p v ┐q LOGICAL EQUIVALENCES ( DISCRETE MATHEMATICS ) pq┐p┐q (p ^ q) ┐(p ^ q) ┐p v ┐q TTFFTFF TFFTFTT FTTFFTT FFTTFTT

15. 15 8. De Morgan’s Laws - b b) ┐(p v q) ≡ ┐p ^ ┐q b) ┐(p v q) ≡ ┐p ^ ┐q LOGICAL EQUIVALENCES ( DISCRETE MATHEMATICS ) pq┐p┐q (p v q) ┐(p v q) ┐p ^ ┐q TTFFTFF TFFTTFF FTTFTFF FFTTFTT

16. 16 9. Absorption Laws - a a) p v (p ^ q) ≡ p a) p v (p ^ q) ≡ p LOGICAL EQUIVALENCES ( DISCRETE MATHEMATICS ) pq (p ^ q) p v (p ^ q) TTTT TFFT FTFF FFFF

17. 17 9. Absorption Laws - b b) p ^ (p v q) ≡ p b) p ^ (p v q) ≡ p LOGICAL EQUIVALENCES ( DISCRETE MATHEMATICS ) pq (p v q) p ^ (p v q) TTTT TFTT FTTF FFFF

18. 18 10. Absorption Laws - a a) p v ┐p ≡ T a) p v ┐p ≡ T LOGICAL EQUIVALENCES ( DISCRETE MATHEMATICS ) p┐p p v ┐p T TFTT FTTT

19. 19 10. Absorption Laws - b b) p ^ ┐p ≡ F b) p ^ ┐p ≡ F LOGICAL EQUIVALENCES ( DISCRETE MATHEMATICS ) p┐p p ^ ┐p F TFFF FTFF

20. THANK YOU! FORHAD AHMED KHAN