Question

Construct a truth table for each of these compound propositions.

a.$\mathbf{(}\mathit{p}\mathbf{\vee }\mathit{q}\mathbf{)}\mathbf{\to }\mathbf{(}\mathit{p}\mathbf{\oplus }\mathit{q}\mathbf{)}$

b.. $\mathbf{(}\mathit{p}\mathbf{\oplus }\mathit{q}\mathbf{)}\mathbf{\to }\mathbf{(}\mathit{p}\mathbf{\wedge }\mathit{q}\mathbf{)}$

c. $\mathbf{(}\mathit{p}\mathbf{\vee }\mathit{q}\mathbf{)}\mathbf{\oplus }\mathbf{(}\mathit{p}\mathbf{\wedge }\mathit{q}\mathbf{)}$

d.$\mathbf{(}\mathit{p}\mathbf{\leftrightarrow }\mathit{q}\mathbf{)}\mathbf{\oplus }\mathbf{(}\mathbf{\neg }\mathit{p}\mathbf{\leftrightarrow }\mathit{q}\mathbf{)}$

e.$\mathbf{(}\mathit{p}\mathbf{\leftrightarrow }\mathit{q}\mathbf{)}\mathbf{\oplus }\mathbf{(}\mathbf{\neg }\mathit{p}\mathbf{\leftrightarrow }\mathbf{\neg }\mathit{r}\mathbf{)}$

f.$\mathbf{(}\mathit{p}\mathbf{\oplus }\mathit{q}\mathbf{)}\mathbf{\to }\mathbf{(}\mathit{p}\mathbf{\oplus }\mathbf{\neg }\mathit{q}\mathbf{)}$

Short answer

a.

p	q	$p\vee q$	$p\oplus q$	$(p\vee q)\to (p\oplus q)$
T	T	T	T	T
T	F	T	F	F
F	T	T	F	F
F	F	F	T	T

b.

p	q	$p\oplus q$	$p\wedge q$	$(p\oplus q)\to (p\wedge q)$
T	T	T	T	T
T	F	F	F	T
F	T	F	F	T
F	F	T	F	F

c.

p	q	$p\vee q$	$p\wedge q$	$(p\vee q)\oplus (p\wedge q)$
T	T	T	T	T
T	F	T	F	F
F	T	T	F	F
F	F	F	F	T

d.

p	q	$\neg p$	$p\leftrightarrow q$	$\neg p\leftrightarrow q$	$(p\leftrightarrow q)\oplus (\neg p\leftrightarrow q)$
T	T	F	T	F	F
T	F	F	F	T	F
F	T	T	F	T	F
F	F	T	T	F	F

e.

p	q	r	$\neg p$	$\neg r$		$\neg p\leftrightarrow \neg r$
T	T	T	F	F	T	T
T	T	F	F	T	T	F
T	F	T	F	F	F	T
T	F	F	F	T	F	F
F	T	T	T	F	F	F
F	T	F	T	T	F	T
F	F	T	T	F	T	F
F	F	F	T	T	T	T

Step-by-step solution

.

...