Counterexamples

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Section 3.3 Counterexamples

If you are asked to disprove a universally quantified statement, it’s enough to exhibit a single counterexample. Indeed, remember that the negation of the statement $\forall x \in D, (p(x) \implies q(x))$ is the statement $\exists x \in D \text{ such that } (p(x) \land(\lnot q(x))\text{.}$

Example 3.3.1.

Disprove: For all $2 \times 2$ matrices $A$ with real entries, if $A$ is not the $2 \times 2$ zero matrix, then $A$ is nonsingular.

It is not the case that every nonzero $2 \times 2$ matrix $A$ with real entries is nonsingular. Indeed, the matrix $A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$ has the property that for any matrix $B$ with entries $b_{i,j}\text{,}$ $AB = \begin{bmatrix} b_{1,1} & 0 \\ 0 & 0 \end{bmatrix}\text{.}$ Therefore, there is no matrix $B$ for which $AB = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\text{,}$ so $A$ is singular.

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