Quiz 3
Discrete Mathematics · 2 problems · solutions hidden, click to reveal
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Problems
Domain: all real numbers. Express the negation of the following without using the negation symbol:
\[\exists x\,((x < -2) \lor (x \ge 5)).\]
\[\exists x\,((x < -2) \lor (x \ge 5)).\]
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\(\neg\exists x((x<-2)\lor(x\ge5)) \equiv \forall x\,\neg((x<-2)\lor(x\ge5)) \equiv \forall x\,(\neg(x<-2)\land\neg(x\ge5))\)
\(\equiv \forall x\,((x \ge -2) \land (x < 5)).\)
\(\equiv \forall x\,((x \ge -2) \land (x < 5)).\)
Domain: all integers. Determine (with justification) the truth value of \(\forall x\,\exists y\,(x = y^2)\).
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Answer: False.
Take \(x = 2\). The statement specializes to \(\exists y\,(2 = y^2)\), but no integer squares to \(2\), so that is false. Hence \(x=2\) is a counterexample and \(\forall x\exists y(x=y^2)\) is false.