Quantifiers: How to Prove/Disprove

To prove a universal statement, we must show that $p(x)$ is true for all the elements in the set. To disprove a universal statement, on the other hand, it is enough to find a single counterexample where $p(x)$ is false.

To prove an existential statement, we only need to find one example where $p(x)$ is true. To disprove an existential statement, on the other hand, we must show that $p(x)$ is false for all the elements the set.

Quantifier	How to Prove	How to Disprove
Universal ($\forall$)	Show $p(x)$ is true for all $x$.	Find a single counterexample where $p(x)$ is false.
Existential ($\exists$)	Find at least one $x$ where $p(x)$ is true.	Show $p(x)$ is false for all $x$ in the set.

Bottom line: Generally, universal statements are harder to prove but easier to disprove. Existential statements are easier to prove but harder to disprove.