Are the following contradictions?

Question

I have the following:

$p\to (q\land p)$

$p\to \neg (q\land p)$

I am asked if they are contradictions, can someone explain what that means exactly.

I did a truth table for both, and if p is True, then both equations are True, does this imply that they are not contradictions? Or, does contradiction simply mean that only one row in the truth table must differ?

Answer 1

No, they are not contradictions because (1) neither statement is a tautology or contradiction in its own right, and (2) the statements' truth values are not diametrically opposed (i.e., they are not opposites).

Answer 2

I did a truth table for both, and if p is True, then both equations are True,

Wrong.   An implication holds true only if the consequent is true whenever the antecedent is.   Yet if $p$ is true then the consequents of the two statements cannot both be true.

The two statements have the following equivalences. $$p\to (q\wedge p) \iff \neg p \vee q \\[2ex] p\to \neg (q\wedge p) \iff \neg p \vee \neg q$$

$$\boxed{\begin{array}{c|c|c|c} p & q & p\wedge q & p\to (p\wedge q) & p\to \neg(p\wedge q) \\ \hline & & & \neg p\vee q & \neg p\vee \neg q \\ \hline T & T & T & T & F \\ T & F & F & F & T \\ F & T & F & T & T \\ F & F & F & T & T \\ \end{array}}$$

However, since it is possible for the two statements to both be true for some $p,q$, then these two statements do not contradict each other.

Two statements contradict each other only if their truth evaluations are opposed for all possible values of their variables.