**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 --- **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** $R$; $S$ --- **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 --- **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 --- **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.