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

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

Prove that $A \cap B = \bar { A } \cup \bar { B }$ giving a proof using logical equivalence.

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

determine whether the set is finite or infinite. If the set is finite, find its size. -{1, 10, 100, 1000, . . .}

describe each sequence recursively. Include initial conditions and assume that the sequences begin with $a _ { 1 }$ . -an = 1 + 2 + 3 + · · · + n

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.

suppose the following are fuzzy sets: F=\{0.7 Ann ,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. - $f : \mathbf { R } \rightarrow \mathbf { R } \text {, where } f ( x ) = 3 x - 5$

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

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

Prove or disprove: For all positive real numbers x and y, $\lfloor x · y\rfloor ≤ \lfloor x \rfloor · \lfloor y \rfloor$ .

determine whether the set is finite or infinite. If the set is finite, find its size. - $\left\{ x \mid x \in \mathbf { Z } \text { and } x ^ { 2 } < 10 \right\}$

Prove or disprove $A \oplus ( B \oplus C ) = ( A \oplus B ) \oplus C$

Suppose f : Z → Z has the rule f(n) = 3n − 1. Determine whether f is onto Z.

determine whether the rule describes a function with the given domain and codomain. - $G : \mathbf { Q } \rightarrow \mathbf { Q } \text {, where } G ( p / q ) = q$

suppose g: A → B and f : B → C where A = {1, 2, 3, 4}, B = {a, b, c}, C = {2, 8, 10}, and g and f are defined by g = f(1, b), (2, a), (3, b), (4, a)g and f = {(a, 8), (b, 10); (c, 2)}. - $\text { Find } f ^ { - 1 }$

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

determine whether the given set is the power set of some set. If the set is a power set, give the set of which it is a power set. - $\{ \emptyset \ , \{ a , \emptyset \} \}$

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

Find the sum 1 − 1/2 + 1/4 − 1/8 + 1/16 − · · · .