discrete_mathematics_with_applications/chapter_2/test_yourself.md

Test Yourself

Page 73

1. An and statement is true when, and only when, both components are _______.

Solution

True.

2. An or statement is false when, and only when, both components are _______.

Solution

False.

3. Two statement forms are logically equivalent when, and only when, they always have _______.

Solution

The same truth values.

4. De Morgan's laws says (1) that the negation of an and statement is logically equivalent to the _______ statement in which each component is _______, and (2) that the negation of an or statement is logically equivalent to the _______ statement in which each component is _______.

Solution

or; negated; and; negated.

5. A tautology is a statement that is always _______.

Solution

true

6. A contradiction is a statement that is always _______.

Solution

false

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Test Yourself

Page 86

1. An if-then statement is false if, and only if, the hypothesis is _______ and the conclusion is _______.

Solution

true; false

2. The negation of "if p then $q$" is _______.

Solution

p and not q.

 p \wedge \neg q 

3. The converse of "if p then $q$" is _______.

Solution

if q then p

 q \to p 

4. The contrapositive of "if p then $q$" is _______.

Solution

if not q then not p.

 \neg q \to \neg p 

5. The inverse of "if p then $q$" is _______.

Solution

if not p then not q.

 \neg p \to \neg q 

6. A conditional statement and its contrapositive are _______.

Solution

logically equivalent.

7. A conditional statement and its converse are not _______.

Solution

logically equivalent.

8. "R is a sufficient condition for $S$" means "if _______ then _______."

Solution

R; S.

9. "R is a necessary condition for $S$" means "if _______ then _______."

Solution

S; R

10. "R only if $S$" means "if _______ then _______."

Solution

R; S

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Test Yourself

Page 99

1. For an argument to be valid means that every argument of the same form whose premises _______ has a _______ conclusion.

are all true; true

2. For an argument to be invalid means that there is an argument of the same form whose premises _______ and whose conclusion _______.

are all true; is false

3. For an argument to be sound means that it is _______ and its premises _______. In this case we can be sure that its conclusion _______.

valid; are all true; is true