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

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

describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1. -1, 101, 10101, 1010101, . . . .

determine whether the rule describes a function with the given domain and codomain. - $f : \mathbf { R } \rightarrow \mathbf { R } \text { where } f ( x ) = \left\{ \begin{array} { l l } x ^ { 2 } & \text { if } x \leq 2 \\x - 1 & \text { if } x \geq 4\end{array} \right.$

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

Suppose $f : \mathbf { Z } \rightarrow \mathbf { Z }$ has the rule $f ( n ) = 3 n ^ { 2 } - 1$ Determine whether $f$ is 1-1.

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

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

Suppose $g : \mathbf { R } \rightarrow \mathbf { R }$ where $g ( x ) = \left\lfloor \frac { x - 1 } { 2 } \right\rfloor$ (a) Draw the graph of g . (b) Is g 1-1? (c) Is g onto R ?

Suppose U={1,2, ... , 9}, A= all multiples of 2, B= all multiples of 3 , and C={3,4,5,6,7} . Find C-(B-A) .

determine whether the rule describes a function with the given domain and codomain. - $g : \mathbf { N } \rightarrow \mathbf { N } \text { where } g ( n ) = \text { any integer } > n$

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

Determine whether $f$ is a function from the set of all bit strings to the set of integers if $f ( S )$ is the position of a 1 bit in the bit string S ,

In questions determine whether the statement is true or false. - $\mathbf { A } \mathbf { B } = \mathbf { B } \mathbf { A } \text { for all } 2 \times 2 \text { matrices } \mathbf { A } \text { and } \mathbf { B } \text {. }$

Suppose $q : A \rightarrow B$ and $f : B \rightarrow C$ where A={1,2,3,4}, B={a, b, c}, C={2,7,10}, and } , and $f$ and g are defind by g={(1, b),(2, a),(3, a),(4, b)} and $f$ ={(a, 10),(b, 7),(c, 2)} . Find $f ^ { - 1 }$ .

determine whether the set is finite or infinite. If the set is finite, find its size. - $\{ 1,10,100,1000 , \ldots \}$

suppose that $g : A \rightarrow B \text { and } f : B \rightarrow C \text { where } A = B = C = \{ 1,2,3,4 \} , g =$ $\{ ( 1,4 ) , ( 2,1 ) , ( 3,1 ) , ( 4,2 ) \} \text {, and } f = \{ ( 1,3 ) , ( 2,2 ) , ( 3,4 ) , ( 4,2 ) \} \text {. }$ - $\text { Find } g \circ ( g \circ g ) \text {. }$ $\text { and } g \text { and } f \text { are defined by } g = \{ ( 1 , b ) , ( 2 , a ) , ( 3 , b ) , ( 4 , a ) \} \text { and } f = \{ ( a , 8 ) , ( b , 10 ) , ( c , 2 ) \} \text {. }$

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

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

suppose $A = \{ 1,2,3,4,5 \} \text {. Mark the statement TRUE or FALSE. }$ - $\{ 2,4 \} \in A \times A$

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