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
  • p ∧ q
  • p (correct)
  • p ∨ q

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

<p>They are never logically equivalent (D)</p> Signup and view all the answers

What is the purpose of De Morgan's Laws?

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

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

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

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

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

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

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

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

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

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

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

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

<p>x ≤ –1 or x &gt; 4 (C)</p> Signup and view all the answers

What is the definition of a tautology?

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

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

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

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

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

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

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

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

<p>Yes (B)</p> Signup and view all the answers

Flashcards

Truth Table Purpose

Visualizes logical relationships between statements

~p ∧ q Truth Table

p is false, q is true

~(~p)

Logical equivalent to p

~(p ∧ q) vs ~p ∧ ~q

Not logically equivalent

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De Morgan's Laws

Convert between conjunctions and disjunctions

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(p ∨ q) ∧ ~(p ∧ q) Truth Table

True when p and q have different truth values

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Negation of 'and'

'or' with negated components

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'Not true that I am not happy'

Logically equivalent to 'I am happy'

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~(p ∧ q)

~p ∨ ~q

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Negation of 'Akram is unfit and Saleem is injured'

Akram is not unfit or Saleem is not injured

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Negation of '-1 < x ≤ 4'

x ≤ –1 or x > 4

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Tautology

Statement form always true

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~(p ∧ (q ∧ r))

~p ∨ ~q ∨ ~r

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~(p ∨ q)

~p ∧ ~q

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~(The fan is slow or it is very hot)

The fan is not slow and it is not very hot

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(p ∧ q) ∧ r vs p ∧ (q ∧ r)

Logically equivalent

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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

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Description

This quiz covers truth tables for various logical statements, including ~p∧q, ~p ∧ (q ∨ ~r), and (p∨q) ∧ ~(p∧q).

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