Math Problem Statement

Complete the following truth tables for logical operations involving negation and conjunction.

Solution

Let's go through the truth tables one by one and solve the questions based on the image.

Part a - Completing the truth table:

You are asked to fill out the truth table for p, ~p, and ~(~p).

p~p~(~p)
TFT
FTF
  • The column for ~p is the negation of p.
  • The column for ~(~p) is the negation of ~p, which makes it equivalent to p.

Part b - Completing the truth table:

This part involves filling in the truth table for logical disjunction () and conjunction () operations:

p~pp ∨ ~pp ∧ ~p
TFTF
FTTF
  • p ∨ ~p: This is the disjunction (OR) between p and ~p. It results in true whenever at least one operand is true.
  • p ∧ ~p: This is the conjunction (AND) between p and ~p. It results in false because p and ~p are always opposites, and both can't be true at the same time.

Part c - Is p logically equivalent to ~(~p)?

Yes, based on the truth table in part (a), p is logically equivalent to ~(~p). Both columns for p and ~(~p) are identical, showing the same truth values.

Part d - Is p ∨ ~p logically equivalent to p ∧ ~p?

No, based on the truth table in part (b), p ∨ ~p is not logically equivalent to p ∧ ~p. The truth values for p ∨ ~p are always true, while the truth values for p ∧ ~p are always false.


Would you like more details or further explanation on any part? Here are five related questions to expand on this topic:

  1. What is the difference between a tautology and a contradiction in logic?
  2. How would the truth table change if we introduce a third logical operator, such as XOR (exclusive OR)?
  3. Can you explain the De Morgan's laws based on these truth tables?
  4. How do truth tables help in proving logical equivalence in Boolean algebra?
  5. How can we use these truth tables to evaluate more complex logical expressions?

Tip: In logic, an expression like p ∨ ~p is a tautology (always true), while p ∧ ~p is a contradiction (always false).

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Math Problem Analysis

Mathematical Concepts

Logic
Boolean Algebra

Formulas

Negation: ~p
Conjunction: p ∧ q
Disjunction: p ∨ q
Double Negation: ~(~p)

Theorems

Law of Double Negation
Law of Excluded Middle
Law of Noncontradiction

Suitable Grade Level

College