Question

Construct a truth table for each of these compound propositions. [6 marks]

a) p ⊕ p b) p ⊕￢p

c) p ⊕￢q d) ￢p ⊕￢q

e) (p ⊕ q) ∨ (p ⊕￢q) f ) (p ⊕ q) ∧ (p ⊕￢q)

Answer 1

Let us construct the trush table for the following compound propositions:

a) $p ⊕ p$

$\begin{array}{||c||c|c|c|c|c||} \hline\hline p & p ⊕p \\ \hline\hline 0 & 0 \\ \hline 1 & 0\\ \hline\hline \end{array}$

b) $p ⊕\neg p$

$\begin{array}{||c||c|c|c|c|c||} \hline\hline p & \neg p & p ⊕ \neg p \\ \hline\hline 0 & 1 & 1 \\ \hline 1 & 0 & 1\\ \hline\hline \end{array}$

c) $p ⊕\neg q$

$\begin{array}{||c|c||c|c|c|c||} \hline\hline p & q & \neg 𝑞 & p ⊕\neg q \\ \hline\hline 0 & 0 & 1 & 1 \\ \hline 0 & 1 & 0 & 0\\ \hline 1 & 0 & 1 & 0\\ \hline 1 & 1 & 0 & 1\\ \hline\hline \end{array}$

d) $\neg p ⊕\neg q$

$\begin{array}{||c|c||c|c|c|c||} \hline\hline p & q & \neg 𝑝& \neg q & \neg p ⊕\neg q\\ \hline\hline 0 & 0 & 1 & 1 & 0\\ \hline 0 & 1 & 1 & 0 & 1\\ \hline 1 & 0 & 0 & 1 & 1\\ \hline 1 & 1 & 0 & 0 & 0\\ \hline\hline \end{array}$

e) $(p ⊕ q) ∨ (p ⊕\neg q)$

$\begin{array}{||c|c||c|c|c|c||} \hline\hline p & q & \neg q & p ⊕\neg q & p ⊕ q & (p ⊕ q) ∨ (p ⊕\neg q) \\ \hline\hline 0 & 0 & 1 & 1 & 0 & 1\\ \hline 0 & 1 & 0 & 0 & 1 & 1\\ \hline 1 & 0 & 1 & 0 & 1 & 1\\ \hline 1 & 1 & 0 & 1 & 0 &1\\ \hline\hline \end{array}$

f ) $(p ⊕ q) \land (p ⊕\neg q)$

$\begin{array}{||c|c||c|c|c|c||} \hline\hline p & q & \neg q & p ⊕\neg q & p ⊕ q & (p ⊕ q) \land (p ⊕\neg q) \\ \hline\hline 0 & 0 & 1 & 1 & 0 & 0\\ \hline 0 & 1 & 0 & 0 & 1 & 0\\ \hline 1 & 0 & 1 & 0 & 1 & 0\\ \hline 1 & 1 & 0 & 1 & 0 &0\\ \hline\hline \end{array}$