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

Adapt the Cantor diagonalization argument to show that the set of positive real numbers less than 1 with decimal representations consisting only of 0s and 1s is uncountable.

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

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

suppose $A = \{ 1,2,3,4,5 \} \text {. Mark the statement TRUE or FALSE. }$ - $\{ \emptyset \} \subseteq \mathcal { P } ( A )$

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.

Let $\mathbf { A } = \left( \begin{array} { c c } 1 & m \\0 & 1\end{array} \right)$ Find Aⁿ where n is a positive integer.

Find the sum 2 + 1/2 + 1/8 + 1/32 + 1/128 + · · · .

Give an example of a function $f : Z \rightarrow Z$ that is 1-1 and not onto $\text { Z }$

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

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

Prove that $\overline { A \cap B } = \bar { A } \cup \bar { B }$ by giving a containment proof (that is, prove that the left side is a subset of the right side and that the right side is a subset of the left side).

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

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

suppose g: A → B and $f : B \rightarrow C \text { where } A = \{ 1,2,3,4 \} , B = \{ a , b , c \} , C = \{ 2,8,10 \} \text {, }$ - $\text { Find } f \circ f ^ { - 1 } \text {. }$

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

suppose $A = \{ a , b , c \} \text { and } B = \{ b , \{ c \} \} \text {. Mark the statement TRUE or FALSE. }$ - $\{ \{ \{ c \} \} \} \subseteq \mathcal { P } ( B )$

Give an example of a function f: N → Z that is onto Z and not 1-1.

describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1. -0.1, 0.11, 0.111, 0.1111, . . . .

Give an example of a function $f : \mathbf { Z } \rightarrow \mathbf { N }$ that is 1-1 and not onto $\mathbf { N }$ .

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