Question 1

for each partial function, determine its domain, codomain, domain of deﬁnition, set of
values for which it is undeﬁned or if it is a total function:
-f: Z × Z → Z where f(m, n) = mn.

Accepted Answer

The answer of for each partial function, determine its domain,...

Question 2

For any function f: A → B , define a new function g: P(A) → P(B) as follows: for every S ⊆ A, g(S) = { f(x) |
x ∈ S   Prove that  f is onto if and only if  g is onto.

Accepted Answer

The answer of For any function f: A → B...

Question 3

for each partial function, determine its domain, codomain, domain of deﬁnition, set of
values for which it is undeﬁned or if it is a total function:
-f: Z × Z → Z where f(m, n) = m − n if m > n.

Accepted Answer

The answer of for each partial function, determine its domain,...

Question 4

for each partial function, determine its domain, codomain, domain of deﬁnition, set of
values for which it is undeﬁned or if it is a total function:
-f: Z × Z → Q where f(m, n) = m/n.

Accepted Answer

The answer of for each partial function, determine its domain,...

Question 5

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 6

ﬁnd the inverse of the function f or else explain why the function has no inverse.
-$$f : \mathbf { Z } \rightarrow \mathbf { Z } \text { where } f ( x ) = \left\{ \begin{array} { l l } 
x - 2 & \text { if } x \geq 5 \
x + 1 & \text { if } x \leq 4
\end{array} \right.$$

Accepted Answer

The answer of ﬁnd the inverse of the function f...

Question 7

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

Accepted Answer

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

Question 8

ﬁnd the inverse of the function f or else explain why the function has no inverse.
-$$f : \mathbf { R } \rightarrow \mathbf { R } \text { where } f ( x ) = \lfloor 2 x \rfloor$$

Accepted Answer

The answer of ﬁnd the inverse of the function f...

Question 9

Suppose f: R → R and g: R → R where g(x) = 2x + 1 and g ◦ f(x) = 2x + 11. Find the rule for f.

Accepted Answer

The answer of Suppose f: R → R and g:...

Question 10

ﬁnd the inverse of the function f or else explain why the function has no inverse.
-$$f : \mathbf { Z } \rightarrow \mathbf { Z } \text { where } f ( x ) = x \bmod 10$$

Accepted Answer

The answer of ﬁnd the inverse of the function f...

Question 11

Suppose g: A → B and f: B → C, where f ◦ g is 1-1 and f is 1-1. Must g be 1-1?

Accepted Answer

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

Question 12

for each partial function, determine its domain, codomain, domain of deﬁnition, set of
values for which it is undeﬁned or if it is a total function:
-f: Z → R where f(n) = 1/n.

Accepted Answer

The answer of for each partial function, determine its domain,...

Question 13

Let f: {1, 2, 3, 4, 5} → {1, 2, 3, 4, 5, 6} be a function.
(a) How many total functions are there?
(b) How many of these functions are one-to-one?

Accepted Answer

The answer of Let f: {1, 2, 3, 4, 5}...

Question 14

suppose g $$\text { : } A \rightarrow B \text { and } f : B \rightarrow C \text { where } A = \{ a , b , c , d \} , B = \{ 1,2,3 \} , C = \{ 2,3,6,8 \}$$ $$\text { and } g \text { and } f \text { are defined by } g = \{ ( a , 2 ) , ( b , 1 ) , ( c , 3 ) , ( d , 2 ) \} \text { and } f = \{ ( 1,8 ) , ( 2,3 ) , ( 3,2 ) \} \text {. }$$
-$$\text { Find } f ^ { - 1 } \text {. }$$

Accepted Answer

The answer of suppose g \[\text { : } A...

Question 15

ﬁnd the inverse of the function f or else explain why the function has no inverse.
-$$f : A \rightarrow B \text { where } A = \{ a , b , c \} , B = \{ 1,2,3 \} \text { and } f = \{ ( a , 2 ) , ( b , 1 ) , ( c , 3 ) \}$$

Accepted Answer

The answer of ﬁnd the inverse of the function f...

Question 16

ﬁnd the inverse of the function f or else explain why the function has no inverse.
-$$f : \mathbf { R } \rightarrow \mathbf { R } \text { where } f ( x ) = 3 x - 5$$

Accepted Answer

The answer of ﬁnd the inverse of the function f...

Question 17

Suppose g: A → B and f: B → C, where f ◦ g is 1-1 and g is 1-1. Must f be 1-1?

Accepted Answer

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

Question 18

suppose g $$\text { : } A \rightarrow B \text { and } f : B \rightarrow C \text { where } A = \{ a , b , c , d \} , B = \{ 1,2,3 \} , C = \{ 2,3,6,8 \}$$ $$\text { and } g \text { and } f \text { are defined by } g = \{ ( a , 2 ) , ( b , 1 ) , ( c , 3 ) , ( d , 2 ) \} \text { and } f = \{ ( 1,8 ) , ( 2,3 ) , ( 3,2 ) \} \text {. }$$
-$$\text { Find } f \circ g \text {. }$$

Accepted Answer

The answer of suppose g \[\text { : } A...

Question 19

for each partial function, determine its domain, codomain, domain of deﬁnition, set of
values for which it is undeﬁned or if it is a total function:
-f: Z → Z where f(n) = 8n/29.

Accepted Answer

The answer of for each partial function, determine its domain,...

Question 20

ﬁnd a formula that generates the following sequence a1, a2, a3 . . . .
-5, 9, 13, 17, 21, . . . .

Accepted Answer

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

Quiz 2: Basic Structures: Sets, Functions, Sequences, Sums, Matrices

for each partial function, determine its domain, codomain, domain of deﬁnition, set of values for which it is undeﬁned or if it is a total function: -f: Z × Z → Z where f(m, n) = mn.

For any function f: A → B , define a new function g: P(A) → P(B) as follows: for every S ⊆ A, g(S) = { f(x) | x ∈ S Prove that f is onto if and only if g is onto.

for each partial function, determine its domain, codomain, domain of deﬁnition, set of values for which it is undeﬁned or if it is a total function: -f: Z × Z → Z where f(m, n) = m − n if m > n.

for each partial function, determine its domain, codomain, domain of deﬁnition, set of values for which it is undeﬁned or if it is a total function: -f: Z × Z → Q where f(m, n) = m/n.

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.

ﬁnd the inverse of the function f or else explain why the function has no inverse. - $f : \mathbf { Z } \rightarrow \mathbf { Z } \text { where } f ( x ) = \left\{ \begin{array} { l l } x - 2 & \text { if } x \geq 5 \\x + 1 & \text { if } x \leq 4\end{array} \right.$

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

ﬁnd the inverse of the function f or else explain why the function has no inverse. - $f : \mathbf { R } \rightarrow \mathbf { R } \text { where } f ( x ) = \lfloor 2 x \rfloor$

Suppose f: R → R and g: R → R where g(x) = 2x + 1 and g ◦ f(x) = 2x + 11. Find the rule for f.

ﬁnd the inverse of the function f or else explain why the function has no inverse. - $f : \mathbf { Z } \rightarrow \mathbf { Z } \text { where } f ( x ) = x \bmod 10$

Suppose g: A → B and f: B → C, where f ◦ g is 1-1 and f is 1-1. Must g be 1-1?

for each partial function, determine its domain, codomain, domain of deﬁnition, set of values for which it is undeﬁned or if it is a total function: -f: Z → R where f(n) = 1/n.

Let f: {1, 2, 3, 4, 5} → {1, 2, 3, 4, 5, 6} be a function. (a) How many total functions are there? (b) How many of these functions are one-to-one?

ﬁnd the inverse of the function f or else explain why the function has no inverse. - $f : A \rightarrow B \text { where } A = \{ a , b , c \} , B = \{ 1,2,3 \} \text { and } f = \{ ( a , 2 ) , ( b , 1 ) , ( c , 3 ) \}$

ﬁnd the inverse of the function f or else explain why the function has no inverse. - $f : \mathbf { R } \rightarrow \mathbf { R } \text { where } f ( x ) = 3 x - 5$

Suppose g: A → B and f: B → C, where f ◦ g is 1-1 and g is 1-1. Must f be 1-1?

for each partial function, determine its domain, codomain, domain of deﬁnition, set of values for which it is undeﬁned or if it is a total function: -f: Z → Z where f(n) = 8n/29.

ﬁnd a formula that generates the following sequence a1, a2, a3 . . . . -5, 9, 13, 17, 21, . . . .

Quiz 2: Basic Structures: Sets, Functions, Sequences, Sums, Matrices

for each partial function, determine its domain, codomain, domain of deﬁnition, set of values for which it is undeﬁned or if it is a total function: -f: Z × Z → Z where f(m, n) = mn.

For any function f: A → B , define a new function g: P(A) → P(B) as follows: for every S ⊆ A, g(S) = { f(x) | x ∈ S Prove that f is onto if and only if g is onto.

for each partial function, determine its domain, codomain, domain of deﬁnition, set of values for which it is undeﬁned or if it is a total function: -f: Z × Z → Z where f(m, n) = m − n if m > n.

for each partial function, determine its domain, codomain, domain of deﬁnition, set of values for which it is undeﬁned or if it is a total function: -f: Z × Z → Q where f(m, n) = m/n.

ﬁnd the inverse of the function f or else explain why the function has no inverse. - f:R→R where f(x)=⌊2x⌋f : \mathbf { R } \rightarrow \mathbf { R } \text { where } f ( x ) = \lfloor 2 x \rfloorf:R→R where f(x)=⌊2x⌋

Suppose f: R → R and g: R → R where g(x) = 2x + 1 and g ◦ f(x) = 2x + 11. Find the rule for f.

ﬁnd the inverse of the function f or else explain why the function has no inverse. - f:Z→Z where f(x)=x mod 10f : \mathbf { Z } \rightarrow \mathbf { Z } \text { where } f ( x ) = x \bmod 10f:Z→Z where f(x)=xmod10

Suppose g: A → B and f: B → C, where f ◦ g is 1-1 and f is 1-1. Must g be 1-1?

for each partial function, determine its domain, codomain, domain of deﬁnition, set of values for which it is undeﬁned or if it is a total function: -f: Z → R where f(n) = 1/n.

Let f: {1, 2, 3, 4, 5} → {1, 2, 3, 4, 5, 6} be a function. (a) How many total functions are there? (b) How many of these functions are one-to-one?

ﬁnd the inverse of the function f or else explain why the function has no inverse. - f:R→R where f(x)=3x−5f : \mathbf { R } \rightarrow \mathbf { R } \text { where } f ( x ) = 3 x - 5f:R→R where f(x)=3x−5

Suppose g: A → B and f: B → C, where f ◦ g is 1-1 and g is 1-1. Must f be 1-1?

for each partial function, determine its domain, codomain, domain of deﬁnition, set of values for which it is undeﬁned or if it is a total function: -f: Z → Z where f(n) = 8n/29.

ﬁnd a formula that generates the following sequence a1, a2, a3 . . . . -5, 9, 13, 17, 21, . . . .

ﬁnd the inverse of the function f or else explain why the function has no inverse. - $f : \mathbf { R } \rightarrow \mathbf { R } \text { where } f ( x ) = \lfloor 2 x \rfloor$

ﬁnd the inverse of the function f or else explain why the function has no inverse. - $f : \mathbf { Z } \rightarrow \mathbf { Z } \text { where } f ( x ) = x \bmod 10$

ﬁnd the inverse of the function f or else explain why the function has no inverse. - $f : \mathbf { R } \rightarrow \mathbf { R } \text { where } f ( x ) = 3 x - 5$