To determine whether the given statement [tex]\((p \rightarrow a) \leftrightarrow (\sim p \vee a)\)[/tex] is a tautology, a contradiction, or a contingency, let's go through the evaluation step-by-step.
### Step 1: Understand the Statements
First, understand the logical components:
- [tex]\( p \rightarrow a \)[/tex] (If [tex]\( p \)[/tex], then [tex]\( a \)[/tex])
- [tex]\( \sim p \)[/tex] (Not [tex]\( p \)[/tex])
- [tex]\( \sim p \vee a \)[/tex] (Not [tex]\( p \)[/tex] or [tex]\( a \)[/tex])
- [tex]\( \leftrightarrow \)[/tex] (Logical equivalence)
### Step 2: Express in Truth Table
To evaluate this, we can use a truth table, but instead, we'll logically simplify to avoid an exhaustive truth table since the evaluation shows it's a special logical form.
### Step 3: Logical Equivalence
We'll analyze the logical equivalence:
The implication [tex]\( p \rightarrow a \)[/tex] can be rewritten as [tex]\( \sim p \vee a \)[/tex] because an implication is true in the same cases when "not [tex]\( p \)[/tex]" or [tex]\( a \)[/tex] is true.
The expression [tex]\((p \rightarrow a)\)[/tex] is logically equivalent to [tex]\((\sim p \vee a)\)[/tex]. Thus, [tex]\( (p \rightarrow a) \leftrightarrow (\sim p \vee a) \)[/tex] simplifies to:
### Step 4: Simplify the Statement
- Since [tex]\( p \rightarrow a \)[/tex] is equivalent to [tex]\( \sim p \vee a \)[/tex], rewriting the original statement:
[tex]\[ (p \rightarrow a) \leftrightarrow (\sim p \vee a) \][/tex]
becomes:
[tex]\[ (\sim p \vee a) \leftrightarrow (\sim p \vee a) \][/tex]
### Step 5: Simplified Expression
- The expression [tex]\( (\sim p \vee a) \leftrightarrow (\sim p \vee a) \)[/tex] should be evaluated.
Since any statement is logically equivalent to itself, this results in a true statement universally, i.e., a tautology. This is because both sides are exactly the same, thereby always resulting in true.
### Conclusion:
Given the analysis, the statement [tex]\((p \rightarrow a) \leftrightarrow (\sim p \vee a)\)[/tex] is always true regardless of the truth values of [tex]\( p \)[/tex] and [tex]\( a \)[/tex]. Hence, it is a tautology.
Thus, the statement pattern [tex]\((p \rightarrow a) \leftrightarrow (\sim p \vee a)\)[/tex] is a Tautology.
