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

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

Suppose A= $\left( \begin{array} { l l l } 1 & 0 & 1 \\0 & 1 & 1 \\1 & 1 & 0\end{array} \right)$ and B= $\left( \begin{array} { l l l } 0 & 1 & 0 \\0 & 1 & 1 \\1 & 0 & 0\end{array} \right)$ . Find (a) the join of A and B . (b) the meet of A and B . (c) the Boolean product of A and B .

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

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

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

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

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 g={(1, b),(2, a),(3, a),(4, b)} and $f$ ={(a, 10),(b, 7),(c, 2)} . Find $\text { f ∘ g }$

Suppose $f : \mathbf { R } \rightarrow \mathbf { R }$ where $f ( x ) = \lfloor x / 2 \rfloor$ (a) If $S = \{ x \mid 1 \leq x \leq 6 \}$ , find f(S) . (b) If T={3,4,5} , find $f ^ { - 1 } ( T )$

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

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

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

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 students studying a programming language, the set of students studying Java.

describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1. - $a _ { n } = 5 ^ { n }$

Let f: {1, 2, 3, 4, 5} → {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?