discrete_mathematics_with_applications/chapter_3/test_yourself.md

Test Yourself

Page 141

1. If P(x) is a predicate with domain D, the truth set of P(x) is denoted _______. We read these symbols out loud as _______.

\{x \in D | P(x)\}; "the set of all x in D such that P(x)."

2. Some ways to express the symbol \forall in words are _______.

for every for all, for any, for each, for arbitrary, given any

3. Some ways to express the symbol \exists in words are _______.

there exists, there exist, there exists at least one, for some, for at least one, we can find a

4. A statement of the form \forall x \in D, Q(x) is true if, and only if, Q(x) is _______ for _______.

true; every x in D.

5. A statement of the form \exists x \in D such that Q(x) is true if, and only if, Q(x) is _______ for _______.

true; at least one x in D.

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Test Yourself

Page 152

1. A negation for "All R have property $S$" is "There is _______ R that _______."

exists at least one; does not have property S.

2. A negation for "Some R have property $S$" is "_______."

"No R have property S."

3. A negation for "For every x, if x has property P then x has property $Q$" is "_______."

"There exists at least one x such that x has property P and x does not have property Q."

4. The converse of "For every x, if x has property P then x has property $Q$" is "_______."

"For every x, if x has property Q then x has property P."

5. The contrapositive of "For every x, if x has property P then x has property $Q$" is "_______."

"For every x, if x does not have property Q, then x does not have property P."

6. The inverse of "For every x, if x has property P then x has property $Q$" is "_______."

"For every x, if x does not have property P, then x does not have property Q."