Determine which of the following statements is a tautology.

[tex]$p \vee p$[/tex]
[tex]$\sim\left(p^{\wedge \sim} p\right)$[/tex]
[tex]$p^{\wedge} p$[/tex]
[tex]$p^{\wedge \sim} p$[/tex]

Analyze each logical statement using truth tables.
Statement ∼ ( p ∧ ∼ p ) is always true.
Other statements are not always true.
The tautology is: ∼ ( p ∧ ∼ p ) ​

Explanation

Understanding the Problem We are given four logical statements and need to determine which one is a tautology. A tautology is a statement that is always true, regardless of the truth values of its variables. The statements are:

p ∨ p

∼ ( p ∧ ∼ p )

p ∧ p

p ∧ ∼ p

Analyzing Statement 1 Let's analyze each statement using truth tables.


Statement 1: p ∨ p



p
p ∨ p



True
True


False
False


p ∨ p is true when p is true and false when p is false. Therefore, it is not a tautology.

Analyzing Statement 2 Statement 2: ∼ ( p ∧ ∼ p )




p
∼ p
p ∧ ∼ p
∼ ( p ∧ ∼ p )



True
False
False
True


False
True
False
True


∼ ( p ∧ ∼ p ) is always true, regardless of the value of p. Therefore, it is a tautology.

Analyzing Statement 3 Statement 3: p ∧ p




p
p ∧ p



True
True


False
False


p ∧ p is true when p is true and false when p is false. Therefore, it is not a tautology.

Analyzing Statement 4 Statement 4: p ∧ ∼ p




p
∼ p
p ∧ ∼ p



True
False
False


False
True
False


p ∧ ∼ p is always false. Therefore, it is not a tautology.

Conclusion Based on the truth tables, only statement 2, ∼ ( p ∧ ∼ p ) , is always true. Therefore, it is the only tautology.

Examples
Tautologies are important in logic and computer science because they represent statements that are always true, regardless of the input. For example, in designing digital circuits, engineers use tautologies to simplify logical expressions and ensure that the circuit always produces the correct output. Similarly, in programming, tautologies can be used to verify the correctness of code and to optimize algorithms.

Answered by GinnyAnswer | 2025-07-03

The only tautology among the provided statements is ∼ ( p ∧ ∼ p ) , which is true for all values of p . Other statements are not always true. Thus, the tautology is ∼ ( p ∧ ∼ p ) .
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Answered by Anonymous | 2025-07-04