Question

The statement pattern \(p\rightarrow (q\rightarrow p)\equiv\)

• A. \(p\rightarrow (p\rightarrow q)\)
• B. \(p\rightarrow (p\wedge q)\)
• C. \(p\rightarrow (p\leftrightarrow q)\)
• D. \(p\rightarrow (p\vee q)\)

Hint:

Rewrite the implication using the rule that a conditional is equivalent to not-first or second, then compare with the options.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
The conditional \(a\rightarrow b\) is equivalent to \(\sim a\vee b\).

Step 2: Simplify the given pattern
\(q\rightarrow p\equiv\sim q\vee p\).
So \(p\rightarrow(q\rightarrow p)\equiv\sim p\vee(\sim q\vee p)\).

Step 3: Reduce
\(\sim p\vee p\) is always true, so the whole pattern is a tautology.

Step 4: Compare the options
\(p\rightarrow(p\vee q)\equiv\sim p\vee p\vee q\), which is also always true.
The other options reduce to \(\sim p\vee q\), which is not a tautology.

Final Answer:
The pattern is a tautology, equivalent to p -> (p v q). \[ \boxed{\text{(D) }p\rightarrow (p\vee q)} \]