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

determine whether the set is finite or infinite. If the set is finite, find its size. - $A \times B \text {, where } A = \{ a , b , c \} \text { and } B = \emptyset$

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

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) = mn.

Suppose $f : \mathbf { N } \rightarrow \mathbf { N }$ has the rule $f ( n ) = 4 n + 1$ . Detremine whether $f$ is into $\mathbf { N }$ .

Find the sum 1/4 + 1/8 + 1/16 + 1/32 +... .

describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1. - $a _ { n } = 1 + 2 + 3 + \cdots + n$

Verify that $a _ { n } = 6$ is a solution to the recurrence relation $a _ { n } = 4 a _ { n - 1 } - 3 a _ { n - 2 }$

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

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 → R where f(n) = 1/n.

Suppose g: A → B and f: B → C, where f ◦ g is 1-1 and f is 1-1. Must g be 1-1?

Suppose $f : \mathbf { N } \rightarrow \mathbf { N }$ has the rule $f ( n ) = 4 n ^ { 2 } + 1$ . Determine whether $f$ is onto $\mathrm { N }$ .

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

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

describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1. -0, 1, 0, 1, 0, 1, . . . .

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 where f(n) = 8n/29.

Find $\sum _ { i = 1 } ^ { 6 } \left( ( - 2 ) ^ { i } - 2 ^ { i } \right)$ .

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

Find A² if A={1, a}

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 → Q where f(m, n) = m/n.

Consider these functions from the set of licensed drivers in the state of New York. Is a function one-to-one if it assigns to a licensed driver his or her (a) birthdate (b) mother's first name (c) drivers license number?