Negations

Negation examples

• \(x\) and \(y\) are both even. (\(\sim (P(x)\wedge Q(x))\))

More examples

The square of every real number is non-negative. (\(\forall x\in\mathbb{R}, x^2\ge 0\)).

There is an integer \(y\) so that \(y^2=20\). (\(\exists y\in\mathbb{Z},y^2=20\))

Still more

For every real number \(x\) there is a real number \(y\) so that \(y^3=x\). (\(\forall x,\exists y, y^3=x\))

Conditionals

• \(P\implies Q\) is equivalent to \(\sim P \vee Q\).
  
• \(\sim (P\implies Q)\) is equivalent to \(P\wedge \sim Q\).

If I own a car, I am from South Dakota.

More examples

For every positive real number \(\epsilon\), there is a positive integer \(M\) for which \(x>M\) implies \(|f(x)-b|<\epsilon\).

Note implicit “for all \(x\)” in the implication.

Negation:

There is a positive real number \(\epsilon\) so that for all positive integers \(M\) there is an \(x>M\) and \(|f(x)-b|\ge \epsilon\).

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