Question 1

mark each statement TRUE or FALSE. Assume that the statement applies to all sets.
-$$A \oplus A = A$$

Accepted Answer

A) True 
 B)False

Question 2

Suppose A is a 6 × 8 matrix, B is an 8 × 5 matrix, and C is a 5 × 9 matrix. Find the number of rows, the number of columns, and the number of entries in A(BC).

Accepted Answer

The answer of Suppose A is a 6 × 8...

Question 3

Find $\bigcap _ { i = 1 } ^ { + \infty } \left( 1 - \frac { 1 } { i } , 1 \right)$

Accepted Answer

The answer of Find \(\bigcap _ { i = 1...

Question 4

ﬁnd a formula that generates the following sequence a1, a2, a3 . . . .
-3, 3, 3, 3, 3, . . .

Accepted Answer

The answer of ﬁnd a formula that generates the following...

Question 5

Prove or disprove: $A - ( B \cap C ) = ( A - B ) \cup ( A - C )$

Accepted Answer

The answer of Prove or disprove: \(A - ( B...

Question 6

ﬁnd a formula that generates the following sequence a1, a2, a3 . . . .
-15, 20, 25, 30, 35, . . .

Accepted Answer

The answer of ﬁnd a formula that generates the following...

Question 7

determine whether the rule describes a function with the given domain and codomain.
-$$F : \mathbf { Z } \rightarrow \mathbf { R } , \text { where } F ( x ) = \frac { 1 } { x ^ { 2 } - 5 }$$

Accepted Answer

The answer of determine whether the rule describes a function...

Question 8

Prove that $A \cap ( B \cup C ) = ( A \cap B ) \cup ( A \cap C )$ by giving a Venn diagram proof.

Accepted Answer

The answer of Prove that \(A \cap ( B \cup...

Question 9

Find a  2 X 2  matrix $\mathbf { A } \neq \left( \begin{array} { l l } 
0 & 0 \
0 & 0
\end{array} \right)$ such that $\mathbf { A } ^ { 2 } = \left( \begin{array} { l l } 
0 & 0 \
0 & 0
\end{array} \right)$

Accepted Answer

The answer of Find a  2 X 2 ...

Question 10

suppose the following are fuzzy sets: $$\begin{array} { l } 
F = \{ 0.7 \text { Ann } , 0.1 \text { Bill } , 0.8 \text { Fran } , 0.3 \text { Olive } , 0.5 \text { Tom } \} , \
R = \{ 0.4 \text { Ann } , 0.9 \text { Bill } , 0.9 \text { Fran } , 0.6 \text { Olive } , 0.7 \text { Tom } \}
\end{array}$$
-$$\text { Find } \bar { F } \text { and } \bar { R } \text {. }$$

Accepted Answer

The answer of suppose the following are fuzzy sets: \[\begin{array}...

Question 11

Suppose B = $\left( \begin{array} { l l } 
6 & 2 \
3 & 1
\end{array} \right)$  and C=$\left( \begin{array} { l l } 
2 & 1 \
0 & 6
\end{array} \right)$ .  Find a matrix A such that  AB =C  or prove that no such matrix exists.

Accepted Answer

The answer of Suppose B = \(\left( \begin{array} { l...

Question 12

mark each statement TRUE or FALSE. Assume that the statement applies to all sets.
-$$\text { If } A \cap C = B \cap C \text {, then } A = B$$

Accepted Answer

A) True 
 B)False

Question 13

determine whether the statement is true or false.
-AB=BA for all 2 × 2 matrices A and B.

Accepted Answer

A) True 
 B)False

Question 14

suppose g: A → B and f : B → C where A = {1, 2, 3, 4}, B = {a, b, c}, C = {2, 8, 10}, and g and f are defined by g = f(1, b), (2, a), (3, b), (4, a)g and f = {(a, 8), (b, 10); (c, 2)}.
-$$\text { Explain why } g ^ { - 1 } \text { is not a function. }$$

Accepted Answer

The answer of suppose g: A → B and f...

Question 15

For the partial function $f : \mathbf { Z } \times \mathbf { Z } \rightarrow \mathbf { R }$ defined by $f ( m , n ) = \frac { 1 } { n ^ { 2 } - m ^ { 2 } }$ , determine its domain, codomain, domain of definition, and set of values for which it is undefined or whether it is a total function.

Accepted Answer

The answer of For the partial function \(f : \mathbf...

Question 16

suppose that g: A → B and f : B → C , where A = B = C = {1, 2, 3, 4}, g =
{(1, 4), (2, 1), (3, 1), (4, 2)}, and f = {(1, 3), (2, 2), (3, 4), (4, 2)}.
-$$\text { Find } g \circ g \text {. }$$

Accepted Answer

The answer of suppose that g: A → B and...

Question 17

determine whether the set is ﬁnite or inﬁnite. If the set is ﬁnite, ﬁnd its size.
-$$\{ 1,3,5,7 , \ldots \}$$

Accepted Answer

The answer of determine whether the set is ﬁnite or...

Question 18

determine whether the set is ﬁnite or inﬁnite. If the set is ﬁnite, ﬁnd its size.
-P(A), where A is the power set of {a, b, c}

Accepted Answer

The answer of determine whether the set is ﬁnite or...

Question 19

determine whether the given set is the power set of some set. If the set is a power set, give the set of which it is a power set.
-$$\{ \emptyset , \{ a \} , \{ \emptyset , a \} \}$$

Accepted Answer

The answer of determine whether the given set is the...

Question 20

Prove or disprove: For all positive real numbers x and y, $\lceil x · y\rceil ≤ \lceil x \rceil · \lceil y \rceil$.

Accepted Answer

The answer of Prove or disprove: For all positive real...

Exam 2: A: Basic Structures: Sets, Functions, Sequences, Sums, Matrices

mark each statement TRUE or FALSE. Assume that the statement applies to all sets. - $A \oplus A = A$

Suppose A is a 6 × 8 matrix, B is an 8 × 5 matrix, and C is a 5 × 9 matrix. Find the number of rows, the number of columns, and the number of entries in A(BC).

Find $\bigcap _ { i = 1 } ^ { + \infty } \left( 1 - \frac { 1 } { i } , 1 \right)$

ﬁnd a formula that generates the following sequence a1, a2, a3 . . . . -3, 3, 3, 3, 3, . . .

Prove or disprove: $A - ( B \cap C ) = ( A - B ) \cup ( A - C )$

ﬁnd a formula that generates the following sequence a1, a2, a3 . . . . -15, 20, 25, 30, 35, . . .

determine whether the rule describes a function with the given domain and codomain. - $F : \mathbf { Z } \rightarrow \mathbf { R } , \text { where } F ( x ) = \frac { 1 } { x ^ { 2 } - 5 }$

Prove that $A \cap ( B \cup C ) = ( A \cap B ) \cup ( A \cap C )$ by giving a Venn diagram proof.

Find a 2 X 2 matrix $\mathbf { A } \neq \left( \begin{array} { l l } 0 & 0 \\0 & 0\end{array} \right)$ such that $\mathbf { A } ^ { 2 } = \left( \begin{array} { l l } 0 & 0 \\0 & 0\end{array} \right)$

Suppose B = $\left( \begin{array} { l l } 6 & 2 \\3 & 1\end{array} \right)$ and C= $\left( \begin{array} { l l } 2 & 1 \\0 & 6\end{array} \right)$ . Find a matrix A such that AB =C or prove that no such matrix exists.

mark each statement TRUE or FALSE. Assume that the statement applies to all sets. - $\text { If } A \cap C = B \cap C \text {, then } A = B$

determine whether the statement is true or false. -AB=BA for all 2 × 2 matrices A and B.

suppose g: A → B and f : B → C where A = {1, 2, 3, 4}, B = {a, b, c}, C = {2, 8, 10}, and g and f are defined by g = f(1, b), (2, a), (3, b), (4, a)g and f = {(a, 8), (b, 10); (c, 2)}. - $\text { Explain why } g ^ { - 1 } \text { is not a function. }$

For the partial function $f : \mathbf { Z } \times \mathbf { Z } \rightarrow \mathbf { R }$ defined by $f ( m , n ) = \frac { 1 } { n ^ { 2 } - m ^ { 2 } }$ , determine its domain, codomain, domain of definition, and set of values for which it is undefined or whether it is a total function.

suppose that g: A → B and f : B → C , where A = B = C = {1, 2, 3, 4}, g = {(1, 4), (2, 1), (3, 1), (4, 2)}, and f = {(1, 3), (2, 2), (3, 4), (4, 2)}. - $\text { Find } g \circ g \text {. }$

determine whether the set is ﬁnite or inﬁnite. If the set is ﬁnite, ﬁnd its size. - $\{ 1,3,5,7 , \ldots \}$

determine whether the set is ﬁnite or inﬁnite. If the set is ﬁnite, ﬁnd its size. -P(A), where A is the power set of {a, b, c}

determine whether the given set is the power set of some set. If the set is a power set, give the set of which it is a power set. - $\{ \emptyset , \{ a \} , \{ \emptyset , a \} \}$

Prove or disprove: For all positive real numbers x and y, $\lceil x · y\rceil ≤ \lceil x \rceil · \lceil y \rceil$ .

Exam 2: A: Basic Structures: Sets, Functions, Sequences, Sums, Matrices

mark each statement TRUE or FALSE. Assume that the statement applies to all sets. - A⊕A=AA \oplus A = AA⊕A=A

Suppose A is a 6 × 8 matrix, B is an 8 × 5 matrix, and C is a 5 × 9 matrix. Find the number of rows, the number of columns, and the number of entries in A(BC).

Find ⋂i=1+∞(1−1i,1)\bigcap _ { i = 1 } ^ { + \infty } \left( 1 - \frac { 1 } { i } , 1 \right)⋂i=1+∞​(1−i1​,1)

ﬁnd a formula that generates the following sequence a1, a2, a3 . . . . -3, 3, 3, 3, 3, . . .

Prove or disprove: A−(B∩C)=(A−B)∪(A−C)A - ( B \cap C ) = ( A - B ) \cup ( A - C )A−(B∩C)=(A−B)∪(A−C)

ﬁnd a formula that generates the following sequence a1, a2, a3 . . . . -15, 20, 25, 30, 35, . . .

determine whether the rule describes a function with the given domain and codomain. - F:Z→R, where F(x)=1x2−5F : \mathbf { Z } \rightarrow \mathbf { R } , \text { where } F ( x ) = \frac { 1 } { x ^ { 2 } - 5 }F:Z→R, where F(x)=x2−51​

Prove that A∩(B∪C)=(A∩B)∪(A∩C)A \cap ( B \cup C ) = ( A \cap B ) \cup ( A \cap C )A∩(B∪C)=(A∩B)∪(A∩C) by giving a Venn diagram proof.

Find a 2 X 2 matrix A≠(0000)\mathbf { A } \neq \left( \begin{array} { l l } 0 & 0 \\0 & 0\end{array} \right)A=(00​00​) such that A2=(0000)\mathbf { A } ^ { 2 } = \left( \begin{array} { l l } 0 & 0 \\0 & 0\end{array} \right)A2=(00​00​)

Suppose B = (6231)\left( \begin{array} { l l } 6 & 2 \\3 & 1\end{array} \right)(63​21​) and C= (2106)\left( \begin{array} { l l } 2 & 1 \\0 & 6\end{array} \right)(20​16​) . Find a matrix A such that AB =C or prove that no such matrix exists.

mark each statement TRUE or FALSE. Assume that the statement applies to all sets. - If A∩C=B∩C, then A=B\text { If } A \cap C = B \cap C \text {, then } A = B If A∩C=B∩C, then A=B

determine whether the statement is true or false. -AB=BA for all 2 × 2 matrices A and B.

suppose that g: A → B and f : B → C , where A = B = C = {1, 2, 3, 4}, g = {(1, 4), (2, 1), (3, 1), (4, 2)}, and f = {(1, 3), (2, 2), (3, 4), (4, 2)}. - Find g∘g. \text { Find } g \circ g \text {. } Find g∘g.

determine whether the set is ﬁnite or inﬁnite. If the set is ﬁnite, ﬁnd its size. - {1,3,5,7,…}\{ 1,3,5,7 , \ldots \}{1,3,5,7,…}

determine whether the set is ﬁnite or inﬁnite. If the set is ﬁnite, ﬁnd its size. -P(A), where A is the power set of {a, b, c}

determine whether the given set is the power set of some set. If the set is a power set, give the set of which it is a power set. - {∅,{a},{∅,a}}\{ \emptyset , \{ a \} , \{ \emptyset , a \} \}{∅,{a},{∅,a}}

Prove or disprove: For all positive real numbers x and y, ⌈x⋅y⌉≤⌈x⌉⋅⌈y⌉\lceil x · y\rceil ≤ \lceil x \rceil · \lceil y \rceil⌈x⋅y⌉≤⌈x⌉⋅⌈y⌉ .

mark each statement TRUE or FALSE. Assume that the statement applies to all sets. - $A \oplus A = A$

Find $\bigcap _ { i = 1 } ^ { + \infty } \left( 1 - \frac { 1 } { i } , 1 \right)$

Prove or disprove: $A - ( B \cap C ) = ( A - B ) \cup ( A - C )$

determine whether the rule describes a function with the given domain and codomain. - $F : \mathbf { Z } \rightarrow \mathbf { R } , \text { where } F ( x ) = \frac { 1 } { x ^ { 2 } - 5 }$

Prove that $A \cap ( B \cup C ) = ( A \cap B ) \cup ( A \cap C )$ by giving a Venn diagram proof.

Find a 2 X 2 matrix $\mathbf { A } \neq \left( \begin{array} { l l } 0 & 0 \\0 & 0\end{array} \right)$ such that $\mathbf { A } ^ { 2 } = \left( \begin{array} { l l } 0 & 0 \\0 & 0\end{array} \right)$

Suppose B = $\left( \begin{array} { l l } 6 & 2 \\3 & 1\end{array} \right)$ and C= $\left( \begin{array} { l l } 2 & 1 \\0 & 6\end{array} \right)$ . Find a matrix A such that AB =C or prove that no such matrix exists.

mark each statement TRUE or FALSE. Assume that the statement applies to all sets. - $\text { If } A \cap C = B \cap C \text {, then } A = B$

suppose that g: A → B and f : B → C , where A = B = C = {1, 2, 3, 4}, g = {(1, 4), (2, 1), (3, 1), (4, 2)}, and f = {(1, 3), (2, 2), (3, 4), (4, 2)}. - $\text { Find } g \circ g \text {. }$

determine whether the set is ﬁnite or inﬁnite. If the set is ﬁnite, ﬁnd its size. - $\{ 1,3,5,7 , \ldots \}$

determine whether the given set is the power set of some set. If the set is a power set, give the set of which it is a power set. - $\{ \emptyset , \{ a \} , \{ \emptyset , a \} \}$

Prove or disprove: For all positive real numbers x and y, $\lceil x · y\rceil ≤ \lceil x \rceil · \lceil y \rceil$ .