I know that the logical negation of $$\neg(a \rightarrow b)= a \wedge \neg b $$ I am not clear what that means in the following simple setting:
So its clear that $$x\geq 2 \to x^2\geq 4.$$ Now I can write the logical negation of $a\to b$ as $a \wedge \neg b$, but what does that intuitively mean?
Suppose I want to prove "$a \wedge \neg b$", what do i need to prove mathematically?
thnks
Best Answer
Say $a$ is $x\ge2$ and $b$ is $x^2\ge14$ (and formally an universal quantifier should be involved as in @JMoravitz comment, i.e. $\forall x, x\ge2\to x^2\ge14$ ). Pick $x=3$, then $a$ is true, but $b$ is false. In other words, $x\ge2$ does not imply that $x^2\ge14$. Formally, the negation here is $\exists x, (x\ge2 \land\neg x^2\ge14)$.
Formally, one may separate syntax from sematic (or form from meaning). Given any statements $a$ and $b$, the negation of the formula $a\to b$ is the formula $a\land\neg b$. If you want to prove that $a\land\neg b$ then you need to either use some previously proven formulas, or axioms (accepted without proof), or to interpret $a$ and $b$ in some known model (as for the reals above), giving each of $a$ and $b$ meaning and truth values.