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

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Test Yourself

Page 113

1. The input/output table for a digital logic circuit is a table that shows _______.

The output signal(s) that correspond to all possible combinations of input signals to the circuit.

2. The Boolean expression that corresponds to a digital logic circuit is _______.

a Boolean expression that represents the input signals as variables and indicates the successive actions of the logic gates on the input signals.

3. A recognizer is a digital logic circuit that _______.

outputs a 1 for exactly one particular combination of input signals and outputs 0s for all other combinations.

4. Two digital logic circuits are equivalent if, and only if, _______.

they have the same input/output table

5. A NAND-gate is constructed by placing a _______ gate immediately following an _______ gate.

NOT; AND

6. A NOR-gate is constructed by placing a _______ gate immediately following an _______ gate.

NOT; OR

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Test Yourself

Page 129

1. To represent a nonnegative integer in binary notation means to write it as a sum of products of the form _______, where _______.

d \cdot 2^n d = 1, and n is a nonnegative integer.

2. To add integers in binary notation, you use the facts that 1_2 + 1_2 = _______ and 1_2 + 1_2 + 1_2 = _______.

10_2; 11_2

3. To subtract integers in binary notation, you use the facts that $10_2 - 1_2 = $ _______ and 11_2 - 1_2 = _______.

1_2; 10_2

4. A half-adder is a digital logic circuit that _______, and a full-adder is a digital logic circuit that _______.

outputs the sum of any two binary digits; outputs the sum of any three binary digits

5. If a is an integer with -128 \leq a \leq 127, the 8-bit two's complement of a is _______ if a \geq 0 and is _______ if a < 0.

The 8-bit representation of a; The 8-bit representation of 2^8 - |a|

6. To find the 8-bit two's complement of a negative integer a that is at least -128, you _______, _______, and _______.

• Write the 8-bit binary representation for |a|.

• Switch all the 1's to 0's and all the 0's to 1's. (This is called flipping, or complementing, the bits.)

• Add 1 in binary notation.

7. To add two integers in the range -128 through 127 whose sum is also in the range -128 through 127, you _______, ______, _______, and _______.

• Convert both integers to their 8-bit two's complement representations.

• Add the resulting integers using ordinary binary addition,

• Discarding any carry bit of 1 that may occur in the 2⁸th position.

• Convert the result back to decimal form.

8. To represent a nonnegative integer in hexadecimal notation means to write it as a sum of products of the form _______, where _______.

d \cdot 16^n; d = 0, 1, 2, \dots 9, A, B, C, D, E, F and n is a nonnegative integer.

9. To convert a nonnegative integer from hexadecimal to binary notation, you _______ and _______.

• Write each hexadecimal digit of the integer in 4-bit binary notation.

• Juxtapose the results.