Let the universal set $U = \mathbb{R}$, set $M = \{ x \mid x < 1 \}$, $N = \{ x \mid - 1 < x < 2 \}$, then $\{ x \mid x \geqslant 2 \} =$
A. $C _ { U } ( M \cup N )$
B. $N \cup C _ { U } M$
C. $C _ { U } ( M \cap N )$
D. $M \cup C _ { U } N$

Let $A=\{0,-a\}$, $B=\{1,a-2,2a-2\}$. If $A\subseteq B$, then $a=$
A. 2
B. 1
C. $\frac{2}{3}$

Given sets $M = \{ x \mid - 4 < x \leq 1 \} , N = \{ x \mid - 1 < x < 3 \}$, then $M \cup N =$ \_\_\_\_

Given sets $A = \left\{ x \mid - 5 < x ^ { 3 } < 5 \right\} , B = \{ - 3 , - 1,0,2,3 \}$ , then $A \cap B =$
A. $\{ - 1,0 \}$
B. $\{ 2,3 \}$
C. $\{ - 3 , - 1,0 \}$
D. $\{ - 1,0,2 \}$

Let the universal set $U = \{x \mid x \text{ is a positive integer less than } 9\}$, and set $A = \{1,3,5\}$. Then the number of elements in $\complement_U A$ is
A. $2$
B. $3$
C. $5$
D. $8$

Given set $A = \{-4, 0, 1, 2, 8\}$, $B = \{x \mid x^3 = x\}$, then $A \cap B = $ ( )
A. $\{0, 1, 2\}$
B. $\{1, 2, 8\}$
C. $\{2, 8\}$
D. $\{0, 1\}$

Let the universal set $U = \{1,2,3,4,5,6,7,8\}$, and set $A = \{1,3,5\}$. The number of elements in $C_U A$ is
A. $0$
B. $3$
C. $5$
D. $8$

Let $A = \{1, 2, 3, 4, 5, 6, 7\}$. Then the relation $R = \{(x, y) \in A \times A : x + y = 7\}$ is
(1) symmetric but neither reflexive nor transitive
(2) an equivalence relation
(3) reflexive but neither symmetric nor transitive
(4) transitive but neither reflexive nor symmetric

Define a relation R on the interval $\left[0, \frac{\pi}{2}\right)$ by $x\mathrm{R}y$ if and only if $\sec^2 x - \tan^2 y = 1$. Then R is:
(1) both reflexive and transitive but not symmetric
(2) an equivalence relation
(3) reflexive but neither symmetric nor transitive
(4) both reflexive and symmetric but not transitive

Consider the sets $A = \{ 4 m \mid m$ is a natural number $\}$ and $B = \{ 6 m \mid m$ is a natural number $\}$.
(1) For each of the following $\mathbf { L } \sim \mathbf{O}$, choose the correct answer from among (0) $\sim$ (3) below.
Let $n$ be a natural number.
(i) $n \in A$ is $\mathbf { L }$ for $n$ to be divisible by 2 .
(ii) $n \in B$ is $\mathbf { M }$ for $n$ to be divisible by 24 .
(iii) $n \in A \cup B$ is $\mathbf { N }$ for $n$ to be divisible by 3 .
(iv) $n \in A \cap B$ is $\square\mathbf{O}$ for $n$ to be divisible by 12 . (0) a necessary and sufficient condition
(1) a necessary condition but not a sufficient condition
(2) a sufficient condition but not a necessary condition
(3) neither a necessary condition nor a sufficient condition
(2) Let $C = \{ m \mid m$ is a natural number satisfying $1 \leqq m \leqq 100 \}$.
The number of elements which belong to $( \bar { A } \cup \bar { B } ) \cap C$ is $\mathbf { P Q }$, and the number of elements which belong to $\bar { A } \cap \bar { B } \cap C$ is $\mathbf { R S }$. Note that $\bar { A }$ and $\bar { B }$ denote the complements of $A$ and $B$, where the universal set is the set of all natural numbers.

Consider the sets $A = \{ 4 m \mid m$ is a natural number $\}$ and $B = \{ 6 m \mid m$ is a natural number $\}$.
(1) For each of the following $\mathbf { L } \sim \square \mathbf { O }$, choose the correct answer from among (0) $\sim$ (3) below.
Let $n$ be a natural number.
(i) $n \in A$ is $\mathbf { L }$ for $n$ to be divisible by 2 .
(ii) $n \in B$ is $\mathbf { M }$ for $n$ to be divisible by 24 .
(iii) $n \in A \cup B$ is $\mathbf { N }$ for $n$ to be divisible by 3 .
(iv) $n \in A \cap B$ is $\mathbf{O}$ for $n$ to be divisible by 12 . (0) a necessary and sufficient condition
(1) a necessary condition but not a sufficient condition
(2) a sufficient condition but not a necessary condition
(3) neither a necessary condition nor a sufficient condition
(2) Let $C = \{ m \mid m$ is a natural number satisfying $1 \leqq m \leqq 100 \}$.
The number of elements which belong to $( \bar { A } \cup \bar { B } ) \cap C$ is $\mathbf { P Q }$, and the number of elements which belong to $\bar { A } \cap \bar { B } \cap C$ is $\mathbf { R S }$. Note that $\bar { A }$ and $\bar { B }$ denote the complements of $A$ and $B$, where the universal set is the set of all natural numbers.

Below is the graph of a function $f$. $( a > 2 , b < 1 )$
Accordingly, which of the following could be the graph of the function $| \mathbf { f } ( \mathbf { x } + \mathbf { 2 } ) | - \mathbf { 1 }$?
A) [graph A]
B) [graph B]
C) [graph C]
D) [graph D]
E) [graph E]

Let m and n be real numbers. In the expansion of
$$\left( \frac { \mathrm { m } } { \mathrm { nx } } + \frac { \mathrm { nx } ^ { 2 } } { \mathrm {~m} } \right) ^ { 3 }$$
arranged according to powers of x, the constant term is 6.
Accordingly, what is the ratio $\frac { m } { n }$?
A) 1
B) 2
C) 3
D) 4
E) 5

Let N be the set of natural numbers. The sets
$$\begin{aligned} & C = \{ 2 n : n \in \mathbb { N } \} \\ & K = \left\{ n ^ { 2 } : n \in \mathbb { N } \right\} \end{aligned}$$
are given. Accordingly, which of the following is an element of the Cartesian product set
$$( \mathrm { K } \backslash \mathrm { C } ) \times ( \mathrm { C } \backslash \mathrm { K } )$$
?
A) $( 3,2 )$
B) $( 9,4 )$
C) $( 15,1 )$
D) $( 16,12 )$
E) $( 25,8 )$

Regarding sets $A$, $B$, and $C$
$$\begin{aligned} & \{ ( 1,2 ) , ( 2,3 ) , ( 3,4 ) \} \subseteq A \times B \\ & \{ ( 1,2 ) , ( 3,4 ) , ( 4,2 ) , ( 4,4 ) \} \subseteq A \times C \end{aligned}$$
it is known that.
Accordingly, I. The set $A \cap B$ has at least 3 elements. II. The set $A \cap C$ has at least 3 elements. III. The set $B \cap C$ has at least 3 elements. which of these statements are always true?
A) Only I
B) Only II
C) Only III
D) I and II
E) I and III

If the number of elements of a set whose all elements are positive integers is one more than the value of the smallest element of this set, this set is called a wide set.
Let $A$, $B$, and $C$ be wide sets,

• $A \cup B \cup C = \{ 1,2,3,4,5,6,7,8,9 \}$
• $A \cap B = \{ 3 \}$
• $1 \in A$
• $6 \in B$

it is known that. Accordingly, which of the following is set $C$?
A) $\{ 1,2 \}$
B) $\{ 3,4,8,9 \}$
C) $\{ 3,5,7,8 \}$
D) $\{ 4,5,6,7,8 \}$
E) $\{ 4,5,7,8,9 \}$