How to simplify this propositional term $(p ⊕ q) \iff p$?

Question

Find a compound proposition that is equivalent to $(p ⊕ q) \iff p$ using only the basic connectives, $∧, ∨, ¬$.

Thank you

Answer 1

So $q$ must be false for equivalency of $p$ and $p \oplus q$ . Hence the answer is simply $\neg q$ !

Answer 2

Iff, equiv, and xnor are the same thing and are all commutative. They are just parity checks of truth. $($equivalent to addition mod $2).$

$$\begin{array} {c} (p \oplus q) \iff p \\ \equiv \\ (p \nLeftrightarrow q) \iff p \\ \equiv \\ \lnot ((p \iff q) \iff p \\ \equiv \\ \lnot (q \iff (p \iff p)) \\ \equiv \\ \lnot (q \iff \top) \\ \equiv \\ \lnot q\\ \end{array}$$

Answer 3

$(p \oplus q) \leftrightarrow p \Leftrightarrow$ (Since $p \oplus q \Leftrightarrow p \leftrightarrow \neg q$)

$(p \leftrightarrow \neg q) \leftrightarrow p \Leftrightarrow$ (Since $\leftrightarrow$ is commutative)

$(\neg q \leftrightarrow p) \leftrightarrow p \Leftrightarrow$ (Since $\leftrightarrow is associative)

$\neg q \leftrightarrow (p \leftrightarrow p) \Leftrightarrow$ (Since $p \leftrightarrow p \Leftrightarrow \top$)

$\neg q \leftrightarrow \top \Leftrightarrow$ (Identity)

$\neg q$