Question

Which of the following statement is a contradiction?

• A. $(p\wedge q)\wedge (\sim (p\vee q\left)\right)$
• B. $p\vee (\sim p\wedge q)$
• C. $(p\to q)\to p$
• D. $\sim p\vee \sim q$

Answer 1

The correct option is A $(p\wedge q)\wedge (\sim (p\vee q\left)\right)$

We check for contradiction for all the given options.

A. $(p\wedge q)\wedge (\sim (p\vee q))$

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

All F so this is a contradiction.

B. $p\vee (\sim p\wedge q)$

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

So not a contradiction.

C. $(p⟶q)\to p$

$p$	$q$	$(p⟶q)$	$(p⟶q)\to p$
T	T	T	T
T	F	F	T
F	T	T	F
F	F	T	F

So it is also not a contradiction.

D. $\sim p\vee \sim q$

$p$	$q$	$\sim p$	$\sim q$	$\sim p\vee \sim q$
T	T	F	F	F
T	F	F	T	T
F	T	T	T	T
F	F	T	T	T

So it is also not a contradiction.