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

determine whether the rule describes a function with the given domain and codomain. - $G : \mathbf { R } \rightarrow \mathbf { R } \text { where } G ( x ) = \left\{ \begin{array} { l l } x + 2 & \text { if } x \geq 0 \\x - 1 & \text { if } x \leq 4\end{array} \right.$

In questions determine whether the statement is true or false. - $\text { If } \mathbf { A } \text { is a } 6 \times 4 \text { matrix and } \mathbf { B } \text { is a } 4 \times 5 \text { matrix, then } \mathbf { A B } \text { has } 16 \text { entries. }$

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

describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1. -1, 111, 11111, 1111111, . . . .

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

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

In questions determine whether the statement is true or false. - $\text { If } \mathbf { A } = \left( \begin{array} { c c } 1 & 3 \\- 5 & 2\end{array} \right) , \text { then } \mathbf { A } ^ { 2 } = \left( \begin{array} { c c } 1 & 9 \\25 & 4\end{array} \right)$

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

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

Suppose A is a 6 X 8 matrix, B is an 8 X 5 matrix, and C is a 5 X 9 matrix. Find the number of rows, the number of columns, and the number of entries in A(BC).

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).

for each partial function, determine its domain, codomain, domain of definition, set of values for which it is undefined or if it is a total function: -f: Z × Z → Z where f(m, n) = m − n if m > n.

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

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

Let $f ( x ) = \left\lfloor x ^ { 3 } / 3 \right\rfloor$ Finf (S) S is: (a) {-2,-1,0,1,2,3} (b) {0,1,2,3,4,5} (c) }{1,5,7,11} . (d) }\{2,6,10,14}

describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1. -The Fibonacci numbers.

Prove that $\overline { A \cap B } = \bar { A } \cup \bar { B }$ by giving an element table proof.

suppose the following are fuzzy sets: F=\{0.7,0.1 Bill, 0.8 Fran, 0.3 Olive, 0.5 Tom \}, R=\{0.4 Ann, 0.9 Bill, 0.9 Fran, 0.6 Olive, 0.7 Tom \}. - $\text { Find } F \cup R \text {. }$

find the inverse of the function f or else explain why the function has no inverse. - $f : \mathbf { R } \rightarrow \mathbf { R } \text { where } f ( x ) = 3 x - 5$

find a recurrence relation with initial condition(s) satisfied by the sequence. Assume a0 is the first term of the sequence. - $a _ { n } = 3 n - 1$