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

mark each statement TRUE or FALSE. Assume that the statement applies to all sets. - $A \oplus A = A$

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

Find $\bigcap _ { i = 1 } ^ { + \infty } \left( 1 - \frac { 1 } { i } , 1 \right)$

find a formula that generates the following sequence a1, a2, a3 . . . . -3, 3, 3, 3, 3, . . .

Prove or disprove: $A - ( B \cap C ) = ( A - B ) \cup ( A - C )$

find a formula that generates the following sequence a1, a2, a3 . . . . -15, 20, 25, 30, 35, . . .

determine whether the rule describes a function with the given domain and codomain. - $F : \mathbf { Z } \rightarrow \mathbf { R } , \text { where } F ( x ) = \frac { 1 } { x ^ { 2 } - 5 }$

Prove that $A \cap ( B \cup C ) = ( A \cap B ) \cup ( A \cap C )$ by giving a Venn diagram proof.

Find a 2 X 2 matrix $\mathbf { A } \neq \left( \begin{array} { l l } 0 & 0 \\0 & 0\end{array} \right)$ such that $\mathbf { A } ^ { 2 } = \left( \begin{array} { l l } 0 & 0 \\0 & 0\end{array} \right)$

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 \} - $\text { Find } \bar { F } \text { and } \bar { R } \text {. }$

Suppose B = $\left( \begin{array} { l l } 6 & 2 \\3 & 1\end{array} \right)$ 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.

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

determine whether the statement is true or false. -AB=BA for all 2 × 2 matrices A and B.

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)}. - $\text { Explain why } g ^ { - 1 } \text { is not a function. }$

For the partial function $f : \mathbf { Z } \times \mathbf { Z } \rightarrow \mathbf { R }$ defined by $f ( m , n ) = \frac { 1 } { n ^ { 2 } - m ^ { 2 } }$ , determine its domain, codomain, domain of definition, and set of values for which it is undefined or whether it is a total function.

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)}. - $\text { Find } g \circ g \text {. }$

determine whether the set is finite or infinite. If the set is finite, find its size. - $\{ 1,3,5,7 , \ldots \}$

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

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. - $\{ \emptyset , \{ a \} , \{ \emptyset , a \} \}$

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