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

Prove that there is no smallest positive rational number.

Prove that between every two rational numbers $a / b \text { and } c / d$ (a) there is a rational number. (b) there are an infinite number of rational numbers.

Suppose f: R → Z where f(x) = 82x − 19. (a) If A = {x | 1 ≤ x ≤ 4} , find f(A). (b) If B={3,4,5,6,7} , find f(B) (c) If C={-9,-8} , find $f ^ { - 1 }$ (C) (d) If D={0.4,0.5,0.6} , find $f ^ { - 1 }$ (D)

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

Find $\sum _ { j = 1 } ^ { 3 } \sum _ { i = 1 } ^ { j } i _ { j }$ .

find a formula that generates the following sequence a1, a2, a3 . . . . -1, 1/3, 1/5, 1/7, 1/9, . . . .

Suppose $f : \mathbf { Z } \rightarrow \mathbf { Z }$ has the rule $f ( n ) = 3 n - 1$ . Determine whether $f$ is onto $\text { Z }$ .

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

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

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

suppose g: A → B and $f : B \rightarrow C \text { where } A = \{ 1,2,3,4 \} , B = \{ a , b , c \} , C = \{ 2,8,10 \} \text {, }$ -Explain why $g ^ { - 1 } \text { is not a function. }$

Determine whether f is a function from the set of all bit strings to the set of integers if f(S) is the largest integer i such that the ith bit of S is 0 and f(S) = 1 when S is the empty string (the string with no bits).

suppose that $g : A \rightarrow B \text { and } f : B \rightarrow C \text { where } A = B = C = \{ 1,2,3,4 \} , g =$ $\{ ( 1,4 ) , ( 2,1 ) , ( 3,1 ) , ( 4,2 ) \} \text {, and } f = \{ ( 1,3 ) , ( 2,2 ) , ( 3,4 ) , ( 4,2 ) \} \text {. }$ - $\text { Find } g \circ g \text {. }$

find the inverse of the function f or else explain why the function has no inverse. - $f : A \rightarrow B \text { where } A = \{ a , b , c \} , B = \{ 1,2,3 \} \text { and } f = \{ ( a , 2 ) , ( b , 1 ) , ( c , 3 ) \}$

Suppose $f : \mathbf { N } \rightarrow \mathbf { N }$ has the rule $f ( n ) = 4 n + 1$ Determine whether $f \text { is } 1 - 1$

suppose $A = \{ 1,2,3,4,5 \} \text {. Mark the statement TRUE or FALSE. }$ - $( 1,1 ) \in A \times A$

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

Prove that $A \cap ( B \cup C ) = ( A \cap B ) \cup ( A \cap C )$ by giving a proof using logical equivalence.

suppose $A = \{ 1,2,3,4,5 \} \text {. Mark the statement TRUE or FALSE. }$ - $\emptyset \subseteq \mathcal { P } ( A )$

Prove that $\overline { A \cap B } = \bar { A } \cup \bar { B }$ by giving a Venn diagram proof.