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Truth Tables Lecture

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Questions and Answers

What is the main purpose of a truth table?

To prove an argument is valid

To test the validity of an argument

To deduce the conclusion of an argument

To visualize the logical relationships between statements (correct)

What can be inferred from the truth table for ~p ∧ q?

p must be false and q must be true (correct)

q must be true and p must be false

p and q can never both be true

p and q must always be true

What is the logical equivalent of ~(~p)?

p ∧ q

p (correct)

p ∨ q

What can be concluded about ~(p ∧ q) and ~p ∧ ~q?

They are never logically equivalent (D) Signup and view all the answers

What is the purpose of De Morgan's Laws?

To convert between conjunctive and disjunctive normal forms (D) Signup and view all the answers

What can be inferred from the truth table for (p ∨ q) ∧ ~(p ∧ q)?

The statement is true when p and q have different truth values (B) Signup and view all the answers

What is the negation of an 'and' statement equivalent to?

An 'or' statement with negated components (A) Signup and view all the answers

What can be concluded about the statement 'It is not true that I am not happy'?

It is logically equivalent to 'I am happy' (B) Signup and view all the answers

What is the symbolic representation of the negation of an 'and' statement?

~(p ∧ q) ≡ ~p ∨ ~q (A) Signup and view all the answers

What is the negation of the statement 'Akram is unfit and Saleem is injured'?

Akram is not unfit or Saleem is not injured (A) Signup and view all the answers

What is the negation of the statement '-1 < x ≤ 4'?

x ≤ –1 or x > 4 (C) Signup and view all the answers

What is the definition of a tautology?

A statement form that is always true regardless of the truth values of the statement variables (C) Signup and view all the answers

What is the negation of the statement 'p ∧ (q ∧ r)'?

~(p ∧ (q ∧ r)) ≡ ~p ∨ ~q ∨ ~r (B) Signup and view all the answers

What is the symbolic representation of the negation of an 'or' statement?

~(p ∨ q) ≡ ~p ∧ ~q (C) Signup and view all the answers

What is the negation of the statement 'The fan is slow or it is very hot'?

The fan is not slow and it is not very hot (D) Signup and view all the answers

Are the statements '(p ∧ q) ∧ r' and 'p ∧ (q ∧ r)' logically equivalent?

Yes (B) Signup and view all the answers

Study Notes

Truth Tables

Truth table for ~p ∧ q:
- ~p ∧ q is false when p is true
- ~p ∧ q is true when p is false and q is true
Truth table for ~p ∧ (q ∨ ~r):
- ~p ∧ (q ∨ ~r) is true when p is false and q is true or r is false
- ~p ∧ (q ∨ ~r) is false when p is true or q is false and r is true
Truth table for (p ∨ q) ∧ ~(p ∧ q):
- (p ∨ q) ∧ ~(p ∧ q) is true when p is true and q is false, or p is false and q is true
- (p ∨ q) ∧ ~(p ∧ q) is false when p and q are both true or both false

Double Negative Property

~(~p) ≡ p (Double Negative Property)
Example: "It is not true that I am not happy" is equivalent to "I am happy"

DeMorgan's Laws

~(p ∧ q) ≡ ~p ∨ ~q (DeMorgan's Law)
~(p ∨ q) ≡ ~p ∧ ~q (DeMorgan's Law)
Example: ~(p ∧ q) is not logically equivalent to ~p ∧ ~q
Application: Give negations for each of the following statements:
- a. The fan is not slow and it is not very hot
- b. Akram is not unfit or Saleem is not injured

Inequalities and DeMorgan's Laws

Use DeMorgan's Laws to write the negation of -1 < x ≤ 4
The negation is: x ≤ –1 or x > 4

Tautology

A tautology is a statement form that is always true regardless of the truth values of the statement variables
Example: p ∨ ~p is a tautology
Truth table for p ∨ ~p:
- p ∨ ~p is always true