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

suppose that g: A → B and f : B → C , where A = B = C = {1, 2, 3, 4}, g = {(1, 4), (2, 1), (3, 1), (4, 2)}, and f = {(1, 3), (2, 2), (3, 4), (4, 2)}. - $\text { Find } g \circ ( g \circ g ) \text {. }$

suppose A = {a, b, c}. Mark the statement TRUE or FALSE. - $\emptyset \subseteq A$

Suppose $f : \mathbf { R } \rightarrow \mathbf { R }$ and $g : \mathbf { R } \rightarrow \mathbf { R }$ where $g ( x ) = 2 x + 1$ and $g \circ f ( x ) = 2 x + 11$ Find the rule for $f$ .

Suppose $g : A \rightarrow B$ and $f : B \rightarrow C$ where A={1,2,3,4} , B={a,b,c} C={2,7,10}, and $f$ and $g$ are defined by $C$ ={(1, b),(2, a),(3, a),(4, b)} and $f$ ={(a, 10),(b, 7),(c, 2)} . Find $f ^ { - 1 }$

Suppose $f : \mathbf { Z } \rightarrow \mathbf { Z }$ has the rule $f ( n ) = 3 n ^ { 2 } - 1$ . Determine whether $f$ is 1-1.

Prove that $\overline { \bar { S } \cup \bar { T } } = S \cap T$ for all sets S and T .

describe each sequence recursively. Include initial conditions and assume that the sequences begin with $a _ { 1 }$ . -a_n = the number of subsets of a set of size n

describe each sequence recursively. Include initial conditions and assume that the sequences begin with $a _ { 1 }$ . -0.1, 0.11, 0.111, 0.1111, . . .

Let $f :$ {1,2,3,4,5} $\rightarrow$ {1,2,3,4,5,6} be a function. (a) How many total functions are there? (b) How many of these functions are one-to-one?

Find $\bigcup _ { i = 1 } ^ { + \infty } [ - 1 / i , 1 / i ]$

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

Prove that $A \cap ( B \cup C ) = ( A \cap B ) \cup ( A \cap C )$ 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 $f : \mathbf { R } \rightarrow \mathbf { R }$ where $f ( x ) = \lfloor x / 2 \rfloor$ . (a) Draw the graph of $f$ . (b) Is $f$ 1-1? (c) Is $f$ onto $\mathbf { R }$ ?

Verify that $a _ { n }$ =6 is a solution to the recurrence relation $a _ { n } = 4 a _ { n - 1 } - 3 a _ { n - 2 }$

Find $\sum _ { i = 1 } ^ { 6 } \left( ( - 2 ) ^ { i } - 2 ^ { i } \right)$

Determine whether f is a function from the set of all bit strings to the set of integers if f(S) is the number of 0 bits in S .

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

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

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

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