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

Find the sum 1/4+1/8+/16+1/32+... .

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 $\mathbf { R }$ ?

suppose the following are fuzzy sets: F=\{0.7 Ann ,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 \} - $\text { Find } F \cap R \text {. }$

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

determine whether the rule describes a function with the given domain and codomain. - $G : \mathbf { R } \rightarrow \mathbf { R } , \text { where } G ( x ) = \left\{ \begin{array} { l l } x + 2 & \text { if } x \geq 0 \\x - 1 & \text { if } x \leq 4\end{array} \right.$

determine whether the rule describes a function with the given domain and codomain. - $h : \mathbf { R } \rightarrow \mathbf { R } \text {, where } h ( x ) = \sqrt { x }$

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. - $\{ \emptyset , \{ a \} , \{ \emptyset \} , \{ a , \emptyset \} \}$

determine whether the statement is true or false. -If A is a 6 × 4 matrix and B is a 4 × 5 matrix, then AB has 16 entries.

use a Venn diagram to determine which relationship, $\subseteq , = , \text { or } \supseteq$ is true for the pair of sets. - $( A - C ) - ( B - C ) , \quad A - B$

suppose A = {x, y} and B = {x, {x}}. Mark the statement TRUE or FALSE. - $| \mathcal { P } ( A ) | = 4$

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

Suppose S={1,2,3,4,5\} . Find $| \mathcal { P } ( S ) |$

suppose the following are multisets: S=\{6\cdota,3\cdotb,2\cdotc,5\cdotd\} T=\{2\cdota,4\cdotb,2\cdotc\} - $\text { Find } S \cup T \text {. }$

Prove that $A \cap ( B \cup C ) = ( A \cap B ) \cup ( A \cap C )$ by giving a proof of using logical equivalence.

use a Venn diagram to determine which relationship, $\subseteq , = , \text { or } \supseteq$ is true for the pair of sets. - $( A - B ) \cup ( A - C ) , \quad A - ( B \cap C )$

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. - $\{ \emptyset , \{ \emptyset \} , \{ a \} , \{ \{ a \} \} , \{ \{ \{ a \} \} \} , \{ \emptyset , a \} , \{ \emptyset , \{ a \} \} , \{ \emptyset , \{ \{ a \} \} \} , \{ a , \{ a \} \} , \{ a , \{ \{ a \} \} \} , \{ \{ a \} , \{ \{ a \} \} \}$ , $\{ \emptyset , a , \{ a \} \} , \{ \emptyset , a , \{ \{ a \} \} \} , \{ \emptyset , \{ a \} , \{ \{ a \} \} \} , \{ a , \{ a \} , \{ \{ a \} \} \} , \{ \emptyset , a , \{ a \} , \{ \{ a \} \} \} \}$

Suppose $g : A \rightarrow B$ and $f : B \rightarrow C$ , where $f \circ g$ is 1-1 and $f$ is 1-1. Must g be 1-1?

Suppose inflation continues at three percent annually. (That is, an item that costs $1.00 now will cost $1.03 next year.) Let an = the value (that is, the purchasing power) of one dollar after n years. (a) Find a recurrence relation for a_n. (b) What is the value of $1.00 after 20 years? (c) What is the value of $1.00 after 80 years? (d) If inflation were to continue at ten percent annually, find the value of $1.00 after 20 years. (e) If inflation were to continue at ten percent annually, find the value of $1.00 after 80 years.

determine whether the set is finite or infinite. If the set is finite, find its size. - $\left\{ x \mid x \in \mathbf { N } \text { and } 9 x ^ { 2 } - 1 = 0 \right\}$

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