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

suppose A = {x, y} and B = {x, {x}}. Mark the statement TRUE or FALSE. - $\{x\} \subseteq A-B$

determine whether the set is finite or infinite. If the set is finite, find its size. -P({a, b, c, d}), where P denotes the power set

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

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

determine whether the set is finite or infinite. If the set is finite, find its size. - $S \times T , \text { where } S = \{ a , b , c \} \text { and } T = \{ 1,2,3,4,5 \}$

Show that $\lceil x\rceil = −\lfloor −x\rfloor$ .

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

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.

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

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

Determine whether f is a function from the set of all bit strings to the set of integers if f(S) is the largest integer i such that the ith bit of S is 0 and f(S) = 1 when S is the empty string (the string with no bits).

find a recurrence relation with initial condition(s) satisfied by the sequence. Assume a0 is the first term of the sequence. - $a _ { n } = 2 ^ { n } + 1$

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

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

Suppose $g : \mathbf { R } \rightarrow \mathbf { R }$ where $g ( x ) = \left\lfloor \frac { x - 1 } { 2 } \right\rfloor$ (a) If S= $\{ x \mid 1 < x < 6 \}$ , find $g ( S)$ (b) If T=\{2\} , find $g ^ { - 1 } ( T )$

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

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 A = {a, b, c} and B = {b, {c}}. Mark the statement TRUE or FALSE. - $| \mathcal { P } ( A \times B ) | = 64$

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

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