16 Questions
What is the main purpose of a truth table?
To visualize the logical relationships between statements
What can be inferred from the truth table for ~p ∧ q?
p must be false and q must be true
What is the logical equivalent of ~(~p)?
p
What can be concluded about ~(p ∧ q) and ~p ∧ ~q?
They are never logically equivalent
What is the purpose of De Morgan's Laws?
To convert between conjunctive and disjunctive normal forms
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
What is the negation of an 'and' statement equivalent to?
An 'or' statement with negated components
What can be concluded about the statement 'It is not true that I am not happy'?
It is logically equivalent to 'I am happy'
What is the symbolic representation of the negation of an 'and' statement?
~(p ∧ q) ≡ ~p ∨ ~q
What is the negation of the statement 'Akram is unfit and Saleem is injured'?
Akram is not unfit or Saleem is not injured
What is the negation of the statement '-1 < x ≤ 4'?
x ≤ –1 or x > 4
What is the definition of a tautology?
A statement form that is always true regardless of the truth values of the statement variables
What is the negation of the statement 'p ∧ (q ∧ r)'?
~(p ∧ (q ∧ r)) ≡ ~p ∨ ~q ∨ ~r
What is the symbolic representation of the negation of an 'or' statement?
~(p ∨ q) ≡ ~p ∧ ~q
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
Are the statements '(p ∧ q) ∧ r' and 'p ∧ (q ∧ r)' logically equivalent?
Yes
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
This quiz covers truth tables for various logical statements, including ~p∧q, ~p ∧ (q ∨ ~r), and (p∨q) ∧ ~(p∧q).
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