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

find a recurrence relation with initial condition(s) satisfied by the sequence. Assume a0 is the first term of the sequence. - $a _ { n } = \sqrt { 2 }$

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

Show that (0, 1) has the same cardinality as (0, 2).

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

find a formula that generates the following sequence a1, a2, a3 . . . . -0, 2, 0, 2, 0, 2, 0, . . . .

For each of the pairs of sets in 1-3 determine whether the first is a subset of the second, the second is a subset of the first, or neither is a subset of the other. -The set of people who were born in the U.S., the set of people who are U.S. citizens.

You take a job that pays $25,000 annually . (a) How much do you earn n years from now if you recive a three percent raise each year ? (b) How much do you earn n years from now if you recive a five percent raise each year ? (c) How much do you earn n years from now if each year you recive a raise of $1000 plus two pwecent of your previous year salary?

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

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

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

Prove or disprove: For all positive real numbers x and y $\lceil x \cdot y \rceil \leq \lceil x \rceil \cdot \lceil y \rceil$

suppose $A = \{ a , b , c \} \text { and } B = \{ b , \{ c \} \} \text {. Mark the statement TRUE or FALSE. }$ - $c \in A - B$

mark each statement TRUE or FALSE. Assume that the statement applies to all sets. - $A \cup ( B \cap C ) = ( A \cup B ) \cap ( A \cup C )$

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

Suppose A = $\left( \begin{array} { l l } 3 & 5 \\2 & 4\end{array} \right)$ and C= $\left( \begin{array} { l l } 2 & 1 \\0 & 6\end{array} \right)$ . Find a matrix B such that AB=C or prove that no such matrix exists.

Prove or disprove: $A - ( B \cap C ) = ( A - B ) \cup ( A - C )$

determine whether the set is finite or infinite. If the set is finite, find its size. - $\left\{ x \mid x \in \mathbf { N } \text { and } 4 x ^ { 2 } - 8 = 0 \right\}$

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

describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1. -1/2, 1/3, 1/4, 1/5, . . . .

describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1. - $1 ^ { 2 } , 2 ^ { 2 } , 3 ^ { 2 } , 4 ^ { 2 } , \ldots$