Question

Construct the truth tables for the following compound propositions.

1. (𝑎 ∧ 𝑏) → (𝑎 ∨ 𝑏)

2. (𝑎 → 𝑏) ∨ (∼ 𝑎 → 𝑏)

3. [(𝑎 → 𝑏) ∧ (𝑎 → 𝑐)] → (𝑎 → (𝑏 ∧ 𝑐))

4. ∼ (𝑏 ∧ 𝑎) ↔ (𝑏 ∨∼ 𝑎)

5. (𝑎 → 𝑏) ∨ 𝑐

Answer 1

1.

$\begin{array}{|c|c|c|c|c|} \hline a & b & a \wedge b & a \vee b & (a \wedge b) \rightarrow(a \vee b) \\ \hline \text { T } & \text { T } & \text { T } & \text { T } & \text { T } \\ \hline \text { T } & \text { F } & \text { F } & \text { T } & \text { T } \\ \hline \text { F } & \text { T } & \text { F } & \text { T } & \text { T } \\ \hline \text { F } & \text { F } & \text { F } & \text { F } & \text { T } \\ \hline \end{array}$

2.

$\begin{array}{|c|c|c|c|c|c|} \hline a & b & a \rightarrow b & \neg a & \neg a \rightarrow b & (a \rightarrow b) \vee(\neg a \rightarrow b) \\ \hline \text { T } & \text { T } & \text { T } & \text { F } & \text { T } & \text { T } \\ \hline \text { T } & \text { F } & \text { F } & \text { F } & \text { T } & \text { T } \\ \hline \text { F } & \text { T } & \text { T } & \text { T } & \text { T } & \text { T } \\ \hline \text { F } & \text { F } & \text { T } & \text { T } & \text { F } & \text { T } \\ \hline \end{array}$

3.

$\begin{array}{|c|c|c|c|c|c|c|c|c|} \hline a & b & c & a \rightarrow b& a \rightarrow c & b \wedge c & a \rightarrow & (a \rightarrow b) \wedge & ((a \rightarrow b) \wedge(a \rightarrow c)) \rightarrow \\ & & & & & & (b \wedge c) & (a \rightarrow c) & (a \rightarrow(b \wedge c)) \\ \hline \text { T } & \text { T } & \text { T } & \text { T } & \text { T } & \text { T } & \text { T } & \text { T } & \text { T } \\ \hline \text { T } & \text { T } & \text { F } & \text { T } & \text { F } & \text { F } & \text { F } & \text { F } & \text { T } \\ \hline \text { T } & \text { F } & \text { T } & \text { F } & \text { T } & \text { F } & \text { F } & \text { F } & \text { T } \\ \hline \text { T } & \text { F } & \text { F } & \text { F } & \text { F } & \text { F } & \text { F } & \text { F } & \text { T } \\ \hline \text { F } & \text { T } & \text { T } & \text { T } & \text { T } & \text { T } & \text { T } & \text { T } & \text { T } \\ \hline \text { F } & \text { T } & \text { F } & \text { T } & \text { T } & \text { F } & \text { T } & \text { T } & \text { T } \\ \hline \text { F } & \text { F } & \text { T } & \text { T } & \text { T } & \text { F } & \text { T } & \text { T } & \text { T } \\ \hline \text { F } & \text { F } & \text { F } & \text { T } & \text { T } & \text { F } & \text { T } & \text { T } & \text { T } \\ \hline \end{array}$

4.

$\begin{array}{|c|c|c|c|c|c|c|} \hline b & a & b \wedge a & \neg a & \neg(b \wedge a) & b \vee \neg a & \neg(b \wedge a) \leftrightarrow(b \vee \neg a) \\ \hline \text { T } & \text { T } & \text { T } & \text { F } & \text { F } & \text { T } & \text { F } \\ \hline \text { T } & \text { F } & \text { F } & \text { T } & \text { T } & \text { T } & \text { T } \\ \hline \text { F } & \text { T } & \text { F } & \text { F } & \text { T } & \text { F } & \text { F } \\ \hline \text { F } & \text { F } & \text { F } & \text { T } & \text { T } & \text { T } & \text { T } \\ \hline \end{array}$

5.

$\begin{array}{|c|c|c|c|c|} \hline a & b & c & a \rightarrow b & (a \rightarrow b) \vee c \\ \hline \text { T } & \text { T } & \text { T } & \text { T } & \text { T } \\ \hline \text { T } & \text { T } & \text { F } & \text { T } & \text { T } \\ \hline \text { T } & \text { F } & \text { T } & \text { F } & \text { T } \\ \hline \text { T } & \text { F } & \text { F } & \text { F } & \text { F } \\ \hline \text { F } & \text { T } & \text { T } & \text { T } & \text { T } \\ \hline \text { F } & \text { T } & \text { F } & \text { T } & \text { T } \\ \hline \text { F } & \text { F } & \text { T } & \text { T } & \text { T } \\ \hline \text { F } & \text { F } & \text { F } & \text { T } & \text { T } \\ \hline \end{array}$