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Beginning Activity Beginning Activity 2: Proving that Statements Are Equivalent
A4US

1.

Let \(X\text{,}\) \(Y\text{,}\) and \(Z\) be statements. Complete a truth table for \(\left[ {\left( {X \to Y} \right) \wedge \left( {Y \to Z} \right)} \right] \to \left( {X \to Z} \right)\text{.}\)

2.

Assume that \(P\text{,}\) \(Q\text{,}\) and \(R\) are statements and that we have proven that the following conditional statements are true:

  • If \(P\) then \(Q\) \(\left( {P \to Q} \right)\text{.}\)

  • If \(Q\) then \(R\) \(\left( {Q \to R} \right)\text{.}\)

  • If \(R\) then \(P\) \(\left( {R \to P} \right)\text{.}\)

Explain why each of the following statements is true.

(a)

\(P\) if and only if \(Q\) \(\left( {P \leftrightarrow Q} \right)\text{.}\)

(b)

\(Q\) if and only if \(R\) \(\left( {Q \leftrightarrow R} \right)\text{.}\)

(c)

\(R\) if and only if \(P\) \(\left( {R \leftrightarrow P} \right)\text{.}\)

Remember that \(X \leftrightarrow Y\) is logically equivalent to \(\left( {X \to Y} \right) \wedge \left( {Y \to X} \right)\text{.}\)