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

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 inflation at three precent annually. (That is , an item that costs $q.00 now will costs $1.03 next year.) ley 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. - $A \times B \text {, where } A = \{ 1,2,3,4,5 \} \text { and } B = \{ 1,2,3 \} \text {. }$

Prove that $\overline { A \cap B } = \bar { A } \cup \bar { B }$ by giving an element table proof.

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

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\}\}\}\}

determine whether the set is finite or infinite. If the set is finite, find its size. - $\left\{ x \mid x \in \mathbf { Z } \text { and } x ^ { 2 } < 10 \right\}$

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

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

In questions determine whether the statement is true or false. - $\text { If } \mathbf { A } \text { and } \mathbf { B } \text { are } 2 \times 2 \text { matrices, then } \mathbf { A } + \mathbf { B } = \mathbf { B } + \mathbf { A } \text {. }$

In questions determine whether the statement is true or false. - $\text { If } \mathbf { A } \text { and } \mathbf { B } \text { are } 2 \times 2 \text { matrices such that } \mathbf { A B } = \left( \begin{array} { l l } 0 & 0 \\0 & 0\end{array} \right) \text {, then } \mathbf { A } = \left( \begin{array} { l l } 0 & 0 \\0 & 0\end{array} \right) \text { or } \mathbf { B } = \left( \begin{array} { l l } 0 & 0 \\0 & 0\end{array} \right) \text {. }$

describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1. - $a _ { n } = n !$

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

determine whether the set is finite or infinite. If the set is finite, find its size. - $\mathcal { P } ( A ) \text {, where } A = \mathcal { P } ( \{ 1,2 \} ) \text {. }$

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

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

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

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.

Verify that $a _ { n } = 3 ^ { n + 4 }$ is a solution to the recurrence relation $a _ { n } = 4 a _ { n - 1 } - 3 a _ { n - 2 }$

determine whether each of the following sets is countable or uncountable. For those that are countably infinite exhibit a one-to-one correspondence between the set of positive integers and that set. -The set of positive rational numbers that can be written with denominators less than 3.