Negation of the Boolean expression p ↔ (q → p) is

To find $~ \left(p ↔ \left(q \to p\right)\right)$

We know $~ \left(p ↔ q\right) ≡ \left(p ∧ ~ q\right) ∨ \left(q ∧ ~ p\right)$

So $~ \left(p ↔ \left(q \to p\right)\right)$

$≡ \left(p ∧ ~ \left(q \to p\right)\right) ∨ \left(\left(q \to p\right) ∧ ~ p\right)$

Now \left(p ∧ ~ \left(q \to p\right)\right) ≡ p ∧ \left(q ∧ ~ p\right) ≡ \left(p ∧ ~ p\right) ∧ q ≡ c$</p><p>and \left(q \to p\right) ∧ ~ p ≡ \left(~ q ∨ p\right) ∧ ~ p ≡ ~ p ∧ \left(~ q ∨ p\right)$

$≡ \left(~ p ∧ ~ q\right) ∨ \left(~ p ∧ p\right) = ~ p ∧ ~ q$

Hence $~ \left(p ↔ \left(q \to p\right)\right) ≡ c ∨ \left(~ p ∧ ~ q\right) ≡ ~ p ∧ ~ q$