Formal Logic - Logical Equivalence Presentation

1. z
FORMAL LOGIC
Discrete Structures I
FOR-IAN V. SANDOVAL

2. z
Lesson 5
LOGICAL EQUIVALENCE

3. z
LEARNING OBJECTIVES
❑ Determine if the logical expression is logically equivalent

4. z
LOGICAL EQUIVALENT
❑ Two statements are said to be logically equivalent (or
equivalent ) if they have the same truth value for every row
of the truth table, that is if x ↔ y is a tautology.
❑ Symbolically, x ≡ y.
❑ i.e.
❑ Show that p ^ (q v r ) and (p ^ q) v (p ^ r ) are equal.

5. z
LOGICAL EQUIVALENT
p q r q v r p ^ (q v r ) p ^ q p ^ r (p ^ q) v (p ^ r )
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F

6. z
LOGICAL EQUIVALENT
p q r q v r p ^ (q v r ) p ^ q p ^ r (p ^ q) v (p ^ r )
T T T T
T T F T
T F T T
T F F F
F T T T
F T F T
F F T T
F F F F

7. z
LOGICAL EQUIVALENT
p q r q v r p ^ (q v r ) p ^ q p ^ r (p ^ q) v (p ^ r )
T T T T T
T T F T T
T F T T T
T F F F F
F T T T F
F T F T F
F F T T F
F F F F F

8. z
LOGICAL EQUIVALENT
p q r q v r p ^ (q v r ) p ^ q p ^ r (p ^ q) v (p ^ r )
T T T T T T
T T F T T T
T F T T T F
T F F F F F
F T T T F F
F T F T F F
F F T T F F
F F F F F F

9. z
LOGICAL EQUIVALENT
p q r q v r p ^ (q v r ) p ^ q p ^ r (p ^ q) v (p ^ r )
T T T T T T T
T T F T T T F
T F T T T F T
T F F F F F F
F T T T F F F
F T F T F F F
F F T T F F F
F F F F F F F

10. z
LOGICAL EQUIVALENT
p q r q v r p ^ (q v r ) p ^ q p ^ r (p ^ q) v (p ^ r )
T T T T T T T T
T T F T T T F T
T F T T T F T T
T F F F F F F F
F T T T F F F F
F T F T F F F F
F F T T F F F F
F F F F F F F F

11. z
LOGICAL EQUIVALENT
❑ Enrichment Exercise
Determine whether the following compound
statements are logically equivalent using truth tables.
1. p →q and ~q →~p
2. p ↔ q and (p →q) ^ (q →p)

12. z
LOGICAL EQUIVALENT
1. p →q and ~q →~p
p q p →q ~q ~p ~q →~p
T T T
T F F
F T T
F F T

13. z
LOGICAL EQUIVALENT
1. p →q and ~q →~p
p q p →q ~q ~p ~q →~p
T T T F
T F F T
F T T F
F F T T

14. z
LOGICAL EQUIVALENT
1. p →q and ~q →~p
p q p →q ~q ~p ~q →~p
T T T F F
T F F T F
F T T F T
F F T T T

15. z
LOGICAL EQUIVALENT
1. p →q and ~q →~p
p q p →q ~q ~p ~q →~p
T T T F F T
T F F T F F
F T T F T T
F F T T T T

16. z
LOGICAL EQUIVALENT
2. p ↔ q and (p →q) ^ (q →p)
p q p ↔ q p →q q →p (p →q) ^ (q →p)
T T T
T F F
F T F
F F T

17. z
LOGICAL EQUIVALENT
2. p ↔ q and (p →q) ^ (q →p)
p q p ↔ q p →q q →p (p →q) ^ (q →p)
T T T T
T F F F
F T F T
F F T T

18. z
LOGICAL EQUIVALENT
2. p ↔ q and (p →q) ^ (q →p)
p q p ↔ q p →q q →p (p →q) ^ (q →p)
T T T T T
T F F F T
F T F T F
F F T T T

19. z
LOGICAL EQUIVALENT
2. p ↔ q and (p →q) ^ (q →p)
p q p ↔ q p →q q →p (p →q) ^ (q →p)
T T T T T T
T F F F T F
F T F T F F
F F T T T T

20. z
LAWS OF LOGICAL EQUIVALENCE
❑ Let p, q, and r stands for any statements.
❑ Let T indicates tautology and F indicates contradiction.
Laws Logical Equivalence
Commutative p ^ q ≡ q ^ p
p v q ≡ q v p
Associative p ^ (q ^ r) ≡ (p ^ q) ^ r
p v (q v r) ≡ (p v q) v r
Distributive p ^ (q v r) ≡ (p ^ q) v (p ^ r)
p v (q ^ r) ≡ (p v q) ^ (p v r)
Identity p ^ T ≡ p
p v F ≡ p
Inverse p ^ ~p ≡ F
p v ~p ≡ T

21. z
LAWS OF LOGICAL EQUIVALENCE
❑ Let p, q, and r stands for any statements.
❑ Let T indicates tautology and F indicates contradiction.
Laws Logical Equivalence
Double Negation ~(~p) ≡ p
Idempotent p ^ p ≡ p
p v p ≡ p
De Morgan’s ~(p ^ q) ≡ ~p v ~q
~(p v q) ≡ ~p ^ ~q
Universal Bound p ^ F ≡ F
p v T ≡ T
Absorption p ^ (p v q) ≡ p
p v (p ^ q) ≡ p

22. z
LAWS OF LOGICAL EQUIVALENCE
❑ Additional logical equivalences are as follows.
Laws Logical Equivalence
Exportation Law (p ^ q) → r ≡ p → (q → r)
Contrapositive p → q ≡ ~q → ~p
Reducto Ad Absurdum p → q ≡ (p ^ ~q) → F
Equivalence p ↔ q ≡ (p → q) ^ ( q → p)
p ↔ q ≡ (~p v q) ^ ( p v ~q)
Implication p → q ≡ ~p v q

23. z
LOGICAL EQUIVALENT EXAMPLE
❑ Simplify the following compound statements using the laws
of equivalence.
1. [p v (~p ^ q)] v (p v ~q)
2. [q v (~p ^ q) v (p v ~q)] ^ ~q

24. z
LOGICAL EQUIVALENT EXAMPLE
1. [p v (~p ^ q)] v (p v ~q)
[p v (~p ^ q)] v (p v ~q) ≡[p v (~p ^ q)] v (p v ~q) ≡ [(p v ~p) ^ (p v q)] v (p v ~q)
Distributive Law
[p v (~p ^ q)] v (p v ~q) ≡ [(p v ~p) ^ (p v q)] v (p v ~q)
Distributive Law
[p v (~p ^ q)] v (p v ~q) ≡ [T ^ (p v q)] v (p v ~q)
Inverse Law
[p v (~p ^ q)] v (p v ~q) ≡ (p v q) v (p v ~q)
Identity Law
[p v (~p ^ q)] v (p v ~q) ≡ (p v q) v (p v ~q)
Identity Law
[p v (~p ^ q)] v (p v ~q) ≡ p v (q v ~q)
Distributive Law
[p v (~p ^ q)] v (p v ~q) ≡ p v (q v ~q)
Distributive Law
[p v (~p ^ q)] v (p v ~q) ≡ p v T
Inverse Law
[p v (~p ^ q)] v (p v ~q) ≡ T
Universal Bound Law

25. z
LOGICAL EQUIVALENT EXAMPLE
2. [q v (p v ~q) v (~ p v ~q)] ^ ~q
[q v (p v ~q) v (~ p v ~q)] ^ ~q ≡[q v (p v ~q) v (~ p v ~q)] ^ ~q ≡ [q v (~q v p) v (~ p v ~q)] ^ ~q
Commutative Law
[q v (p v ~q) v (~ p v ~q)] ^ ~q ≡ [q v (~q v p) v (~ p v ~q)] ^ ~q
Commutative Law
[q v (p v ~q) v (~ p v ~q)] ^ ~q ≡ [(q v ~q) v p v (~ p v ~q)] ^ ~q
Associative Law
[q v (p v ~q) v (~ p v ~q)] ^ ~q ≡ [(q v ~q) v p v (~ p v ~q)] ^ ~q
Associative Law
[q v (p v ~q) v (~ p v ~q)] ^ ~q ≡ [T v p v (~ p v ~q)] ^ ~q
Inverse Law
[q v (p v ~q) v (~ p v ~q)] ^ ~q ≡ [T v p v (~ p v ~q)] ^ ~q
Inverse Law
[q v (p v ~q) v (~ p v ~q)] ^ ~q ≡ [(T v p) v (~ p v ~q)] ^ ~q
Associative Law
[q v (p v ~q) v (~ p v ~q)] ^ ~q ≡ [(p v T) v (~ p v ~q)] ^ ~q
Commutative Law

26. z
LOGICAL EQUIVALENT EXAMPLE
2. [q v (p v ~q) v (~ p v ~q)] ^ ~q
[q v (p v ~q) v (~ p v ~q)] ^ ~q ≡ [T v (~ p v ~q)] ^ ~q
Universal Bound Law
[q v (p v ~q) v (~ p v ~q)] ^ ~q ≡ [T v (~ p v ~q)] ^ ~q
[q v (p v ~q) v (~ p v ~q)] ^ ~q ≡ [(~ p v ~q) v T] ^ ~q
Commutative Law
[q v (p v ~q) v (~ p v ~q)] ^ ~q ≡ T ^ ~q
Universal Bound Law
[q v (p v ~q) v (~ p v ~q)] ^ ~q ≡ T ^ ~q
Universal Bound Law
[q v (p v ~q) v (~ p v ~q)] ^ ~q ≡ ~q ^ T
Commutative Law
[q v (p v ~q) v (~ p v ~q)] ^ ~q ≡ ~q
Identity Law

27. z
LOGICAL EQUIVALENT EXAMPLE
2. [q v (p v ~q) v (~ p v ~q)] ^ ~q
[q v (p v ~q) v (~ p v ~q)] ^ ~q ≡ [q v (~q v p) v (~ p v ~q)] ^ ~q
Commutative Law
[q v (p v ~q) v (~ p v ~q)] ^ ~q ≡ [(q v ~q) v p) v (~ p v ~q)] ^ ~q
Associative Law
[q v (p v ~q) v (~ p v ~q)] ^ ~q ≡ [(q v ~q) v p) v (~ p v ~q)] ^ ~q
Inverse Law
[q v (p v ~q) v (~ p v ~q)] ^ ~q ≡ [(T v p) v (~ p v ~q)] ^ ~q
Associative Law
[q v (p v ~q) v (~ p v ~q)] ^ ~q ≡ [T v p v (~ p v ~q)] ^ ~q
Inverse Law

28. z
LOGICAL EQUIVALENT EXAMPLE
2. [q v (p v ~q) v (~ p v ~q)] ^ ~q
[q v (p v ~q) v (~ p v ~q)] ^ ~q ≡ [T v (~ p v ~q)] ^ ~q
Universal Bound Law
[q v (p v ~q) v (~ p v ~q)] ^ ~q ≡ ~q
Identity Law
[q v (p v ~q) v (~ p v ~q)] ^ ~q ≡ [T v (~ p v ~q)] ^ ~q
Universal Bound Law
[q v (p v ~q) v (~ p v ~q)] ^ ~q ≡ T ^ ~q
Universal Bound Law

29. z
Group Enrichment Exercises
❑ Simplify the following compound statements using the laws
of equivalence.
1. [(p ^ r) v (q ^ r)] v ~q
2. [p v (~p v q) v (p v ~q)] ^ ~q
3. ~(p → q) ^ (p ↔ q)

30. z
Group Enrichment Exercises
1. [(p ^ r) v (q ^ r)] v ~q
[(p ^ r) v (q ^ r)] v ~q ≡
[(p ^ r) v (q ^ r)] v ~q ≡ [(p v q) v ~q] ^ (r v ~q)
Distributive Law
[(p ^ r) v (q ^ r)] v ~q ≡ [(p v (q v ~q)] ^ (r v ~q)
Associative Law
[(p ^ r) v (q ^ r)] v ~q ≡ [(p v q) ^ r] v ~q
Distributive Law
[(p ^ r) v (q ^ r)] v ~q ≡ [(p v q) ^ r] v ~q
Distributive Law
[(p ^ r) v (q ^ r)] v ~q ≡ [(p v q) v ~q] ^ (r v ~q)
Distributive Law
[(p ^ r) v (q ^ r)] v ~q ≡ [(p v (q v ~q)] ^ (r v ~q)
Associative Law
[(p ^ r) v (q ^ r)] v ~q ≡ [(p v T)] ^ (r v ~q)
Inverse Law
[(p ^ r) v (q ^ r)] v ~q ≡ [(p v T)] ^ (r v ~q)
Inverse Law
[(p ^ r) v (q ^ r)] v ~q ≡ T ^ (r v ~q)
Universal Bound Law
[(p ^ r) v (q ^ r)] v ~q ≡ r v ~q
Identity Law
[(p ^ r) v (q ^ r)] v ~q ≡ r v ~q
Identity Law

31. z
Group Enrichment Exercises
1. [(p ^ r) v (q ^ r)] v ~q
[(p ^ r) v (q ^ r)] v ~q ≡ ~q v r
Commutative Law

32. z
Group Enrichment Exercises
2. [p v (~p v q) v (p v ~q)] ^ ~q
[p v (~p v q) v (p v ~q)] ^ ~q ≡[p v (~p v q) v (p v ~q)] ^ ~q ≡[p v (~p v q) v (p v ~q)] ^ ~q ≡ [(p v ~p) v q v (p v ~q)] ^ ~q
Associative Law
[p v (~p v q) v (p v ~q)] ^ ~q ≡ [(p v ~p) v q v (p v ~q)] ^ ~q
Associative Law
[p v (~p v q) v (p v ~q)] ^ ~q ≡ [T v q v (p v ~q)] ^ ~q
Inverse Law
[p v (~p v q) v (p v ~q)] ^ ~q ≡ [T v q v (p v ~q)] ^ ~q
Inverse Law
[p v (~p v q) v (p v ~q)] ^ ~q ≡ [(T v q) v (p v ~q)] ^ ~q
Associative Law
[p v (~p v q) v (p v ~q)] ^ ~q ≡ [(q v T) v (p v ~q)] ^ ~q
Commutative Law
[p v (~p v q) v (p v ~q)] ^ ~q ≡ [T v (p v ~q)] ^ ~q
Universal Bound Law
[p v (~p v q) v (p v ~q)] ^ ~q ≡ [T v (p v ~q)] ^ ~q
Universal Bound Law
[p v (~p v q) v (p v ~q)] ^ ~q ≡ [(p v ~q) v T] ^ ~q
Commutative Law
[p v (~p v q) v (p v ~q)] ^ ~q ≡ T ^ ~q
Universal Bound Law

33. z
Group Enrichment Exercises
2. [p v (~p v q) v (p v ~q)] ^ ~q
[p v (~p v q) v (p v ~q)] ^ ~q ≡ ~q
Identity Law

34. z
Group Enrichment Exercises
3. ~(p → q) ^ (p ↔ q)
~(p → q) ^ (p ↔ q) ≡~(p → q) ^ (p ↔ q) ≡~(p → q) ^ (p ↔ q) ≡ ~(~p v q) ^ (p ↔ q)
Implication Law
~(p → q) ^ (p ↔ q) ≡ ~(~p v q) ^ (p ↔ q)
Implication Law
~(p → q) ^ (p ↔ q) ≡ [~(~p) ^ ~q)] ^ (p ↔ q)
De Morgan’s Law
~(p → q) ^ (p ↔ q) ≡ [~(~p) ^ ~q)] ^ (p ↔ q)
De Morgan’s Law
~(p → q) ^ (p ↔ q) ≡ (p ^ ~q) ^ (p ↔ q)
Double Negation Law
~(p → q) ^ (p ↔ q) ≡ (p ^ ~q) ^ (p ↔ q)
Double Negation Law
~(p → q) ^ (p ↔ q) ≡ (p ^ ~q) ^ [(~p v q) ^ (p v ~q)]
Equivalence Law
~(p → q) ^ (p ↔ q) ≡ (p ^ ~q) ^ (p v ~q) ^ (~p v q)
Commutative Law
~(p → q) ^ (p ↔ q) ≡ (p ^ ~q) ^ (p v ~q) ^ (~p v q)
Commutative Law
~(p → q) ^ (p ↔ q) ≡ [(p ^ ~q) ^ (p v ~q)] ^ (~p v q)
Commutative Law
~(p → q) ^ (p ↔ q) ≡ {(p ^ [~q ^ (p v ~q)]} ^ (~p v q)
Associative Law
~(p → q) ^ (p ↔ q) ≡ {(p ^ [~q ^ (p v ~q)]} ^ (~p v q)
Associative Law

35. z
Group Enrichment Exercises
3. ~(p → q) ^ (p ↔ q)
~(p → q) ^ (p ↔ q) ≡ {(p ^ [~q ^ (~q v p)]} ^ (~p v q)
Commutative Law
~(p → q) ^ (p ↔ q) ≡ {(p ^ [~q ^ (~q v p)]} ^ (~p v q)
Commutative Law
~(p → q) ^ (p ↔ q) ≡ (p ^ ~q) ^ (~p v q)
Absorption Law
~(p → q) ^ (p ↔ q) ≡ (p ^ ~q) ^ (~p v q)
Absorption Law
~(p → q) ^ (p ↔ q) ≡ p ^ [~q ^ (~p v q)]
Associative Law
~(p → q) ^ (p ↔ q) ≡ p ^ [(~q ^ ~p) v (~q ^ q)]
Distributive Law
~(p → q) ^ (p ↔ q) ≡ p ^ [(~q ^ ~p) v (~q ^ q)]
Distributive Law
~(p → q) ^ (p ↔ q) ≡ p ^ [(~q ^ ~p) v (q ^ ~q)]
Commutative Law
~(p → q) ^ (p ↔ q) ≡ p ^ [(~q ^ ~p) v F]
Inverse Law
~(p → q) ^ (p ↔ q) ≡ p ^ (~q ^ ~p)
Identity Law
~(p → q) ^ (p ↔ q) ≡ p ^ (~q ^ ~p)
Identity Law
~(p → q) ^ (p ↔ q) ≡ p ^ [(~q ^ ~p) v F]
Inverse Law

36. z
Group Enrichment Exercises
3. ~(p → q) ^ (p ↔ q)
~(p → q) ^ (p ↔ q) ≡ p ^ (~p ^ ~q )
Commutative Law
~(p → q) ^ (p ↔ q) ≡ p ^ (~p ^ ~q )
Commutative Law
~(p → q) ^ (p ↔ q) ≡ (p ^ ~p) ^ ~q
Associate Law
~(p → q) ^ (p ↔ q) ≡ F ^ ~q
Inverse Law
~(p → q) ^ (p ↔ q) ≡ ~q ^ F
Commutative Law
~(p → q) ^ (p ↔ q) ≡ F
Universal Bound Law

37. z
• Levin, O. (2019). Discrete Mathematics: An Open Introduction 3rd Edition. Colorado: School of Mathematics Science
University of Colorado.
• Aslam, A. (2016). Proposition in Discrete Mathematics retrieved from https://www.slideshare.net/AdilAslam4/chapter-1-
propositions-in-discrete-mathematics
• Operator Precedence retrieved from http://intrologic.stanford.edu/glossary/operator_precedence.html
REFERENCES