when does $a\in\mathbb{R}$ does $\neg(a\leq 15\implies a>1)$ hold?

Question

How can I formally write down for which $a\in\mathbb{R}$ the statement $\neg(a\leq 15\implies a>1)$ holds?

Answer 1

Option #1:

• $\neg(a\leq15\implies{a>1})\iff$

• $\neg(\neg(a\leq15)\vee(a>1))\iff$

• $(a\leq15)\wedge\neg(a>1)\iff$

• $(a\leq15)\wedge(a\leq1)\implies{a\leq1}$

Option #2:

• $\neg(a\leq15\implies{a>1})\iff$

• $\neg(\neg(a\leq15)\vee(a>1))\iff$

• $\neg((a>15)\vee(a>1))\implies$

• $\neg(a>1)\iff{a\leq1}$

Answer 2

Hint: An implication is false precisely when the hypothesis is true and the conclusion is false.