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

Suppose $f : \mathbf { R } \rightarrow \mathbf { R }$ where $f ( x ) = \lceil 2x−1 \rceil$ (a) Draw the graph of $f \text {. }$ (b) Is $f$ 1-1 ?(Explain) (c) Is $f$ onto $z$ ? (Explain)

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

describe each sequence recursively. Include initial conditions and assume that the sequences begin with $a _ { 1 }$ . -1, 111, 11111, 1111111, . . .

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

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

determine whether each of the following sets is countable or uncountable. For those that are countably infinite exhibit a one-to-one correspondence between the set of positive integers and that set. - $\text { The set of irrational numbers between } \sqrt { 2 } \text { and } \pi / 2 \text {. }$

Suppose f : N → N has the rule f(n) = 4n + 1. Determine whether f is onto N.

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

determine whether the rule describes a function with the given domain and codomain. - $f : \mathbf { N } \rightarrow \mathbf { N } , \text { where } f ( n ) = \sqrt { n }$

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

Give an example of a function f : Z → N that is both 1-1 and onto N.

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)}. - $\text { Find } f \circ q$

determine whether the statement is true or false. - $\text { If } \mathbf { A } \text { and } \mathbf { B } \text { are } 2 \times 2 \text { matrices such that } \mathbf { A B } = \left( \begin{array} { l l } 0 & 0 \\0 & 0\end{array} \right) \text {, then } \mathbf { A } = \left( \begin{array} { l l } 0 & 0 \\0 & 0\end{array} \right) \text { or } \mathbf { B } = \left( \begin{array} { l l } 0 & 0 \\0 & 0\end{array} \right)$

For any function $f : A \rightarrow B$ , define a new function $g : \mathcal { P } ( A ) \rightarrow \mathcal { P } ( B )$ as follows : for every $S \subseteq A , g ( S )$ = $\{ f ( x ) \mid x \in S \}$ Prove that $f$ is onto if and only if $q$ is onto.

Suppose U = {1, 2, . . . , 9}, A = all multiples of 2, B = all multiples of 3, and C = {3, 4, 5, 6, 7}. Find C − (B − A).

describe each sequence recursively. Include initial conditions and assume that the sequences begin with $a _ { 1 }$ . -0, 1, 0, 1, 0, 1, . . .

find a recurrence relation with initial condition(s) satisfied by the sequence. Assume a0 is the first term of the sequence. - $a _ { n } = \sqrt { 2 }$

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: - $f : \mathbf { Z } \times \mathbf { Z } \rightarrow \mathbf { Z } , \text { where } f ( m , n ) = m n$

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: - $f : \mathbf { Z } \rightarrow \mathbf { R } , \text { where } f ( n ) = 1 / n$

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).