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1. To prove a statement by contradiction, you suppose that ______ and you show 1. To prove a statement by contradiction, you suppose that ______ and you show
that ______. that ______.
the statement is false; this supposition leads to a contradiction
2. A proof by contraposition of a statement of the form 2. A proof by contraposition of a statement of the form
"$\forall x \in D, \text{ if } P(x) \text{ then } Q(x)$" is a direct proof of "$\forall x \in D, \text{ if } P(x) \text{ then } Q(x)$" is a direct proof of
______. ______.
$\forall x \in D, \text{ if } \neg Q(x) \text{ then } \neg P(x)$
3. To prove a statement of the form 3. To prove a statement of the form
"$\forall x \in D, \text{ if } P(x) \text{ then } Q(x)$" by contraposition, "$\forall x \in D, \text{ if } P(x) \text{ then } Q(x)$" by contraposition,
you suppose that ______ and you show that ______. you suppose that ______ and you show that ______.
$Q(x)$ is false; $P(x)$ is false.