14. From the subsets of the set $U = \{ a , b , c , d \}$, select 2 different subsets that must satisfy both of the following conditions:
(1) Both $a$ and $b$ must be selected;
(2) For any two selected subsets $A$ and $B$, we must have $A \subseteq B$ or $B \subseteq A$. Then there are $\_\_\_\_$ $36$ different ways.

Analysis: By enumeration, there are 36 ways in total.

II. Multiple Choice Questions (Total Score: 20 points) This section contains 4 questions. Each question has exactly one correct answer. Candidates must shade the box corresponding to the correct answer on the answer sheet. Each correct answer is worth 5 points; otherwise, zero points are awarded.
15. ``$x = 2 k \pi + \frac { \pi } { 4 } ( k \in \mathbb{Z} )$'' is a \_\_\_\_ condition for ``$\tan x = 1$''. [Answer] (A)
(A) Sufficient but not necessary condition.
(B) Necessary but not sufficient condition.
(C) Sufficient condition.
(D) Neither sufficient nor necessary condition.
Analysis: $\tan \left( 2 k \pi + \frac { \pi } { 4 } \right) = \tan \frac { \pi } { 4 } = 1$, so it is sufficient; However, the converse does not hold. For example, $\tan \frac { 5 \pi } { 4 } = 1$, so it is not necessary.

9. Let $A = \{(x,y) \mid x^2 + y^2 \leq 1, x, y \in \mathbf{Z}\}$, $B = \{(x,y) \mid |x| \leq 2, |y| \leq 2, x, y \in \mathbf{Z}\}$. Define $A \oplus B = \{(x_1 + x_2, y_1 + y_2) \mid (x_1, y_1) \in A, (x_2, y_2) \in B\}$. The number of elements in $A \oplus B$ is
A. 77
B. 49
C. 45
D. 30

146- If $A = \{2k-1 \mid k \in \mathbb{Z},\, 1 \leq k \leq 5\}$ and $B = \{k \in \mathbb{Z} : |k-3| \leq 2\}$, then the set $(A \times B) \cap (B \times A)$ has how many elements?
(1) $6$ (2) $8$ (3) $9$ (4) $16$
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144- If $A = \{x \in \mathbb{N},\ 5 < x^2 < 50\}$ and $B = \{3k-2 \mid k \in \mathbb{Z},\ 1 \leq k \leq 4\}$, then the number of elements of $(A \times B) \cap (B \times A)$ is:
(1) $4$ (2) $8$ (3) $16$ (4) $32$

Let $S = \{ 1, 2, \ldots, n \}$. The number of possible pairs of the form $(A, B)$ with $A \subseteq B$ for subsets $A$ and $B$ of $S$ is
(A) $2^n$
(B) $3^n$
(C) $\sum _ { k = 0 } ^ { n } \binom { n } { k } \binom { n } { n - k }$
(D) $n!$

Let $S = \{ 1,2 , \ldots , n \}$. Find the number of unordered pairs $\{ A , B \}$ of subsets of $S$ such that $A$ and $B$ are disjoint, where $A$ or $B$ or both may be empty.

Let $S = \{ 1,2 , \ldots , n \}$. Find the number of unordered pairs $\{ A , B \}$ of subsets of $S$ such that $A$ and $B$ are disjoint, where $A$ or $B$ or both may be empty.

Let $S = \{ 1, 2, \ldots, n \}$. The number of possible pairs of the form $(A, B)$ with $A \subseteq B$ for subsets $A$ and $B$ of $S$ is
(A) $2 ^ { n }$
(B) $3 ^ { n }$
(C) $\sum _ { k = 0 } ^ { n } \binom { n } { k } \binom { n } { n - k }$
(D) $n !$

Let $S = \{ 1, 2, \ldots , n \}$. The number of possible pairs of the form $(A, B)$ with $A \subseteq B$ for subsets $A$ and $B$ of $S$ is
(A) $2 ^ { n }$
(B) $3 ^ { n }$
(C) $\sum _ { k = 0 } ^ { n } \binom { n } { k } \binom { n } { n - k }$
(D) $n !$

The number of subsets of $\{ 1,2,3 , \ldots , 10 \}$ having an odd number of elements is
(A) 1024
(B) 512
(C) 256
(D) 50 .

Let $S = \{ 1,2 , \ldots , n \}$. For any non-empty subset $A$ of $S$, let $l ( A )$ denote the largest number in $A$. If $f ( n ) = \sum _ { A \subseteq S } l ( A )$, that is, $f ( n )$ is the sum of the numbers $l ( A )$ while $A$ ranges over all the nonempty subsets of $S$, then $f ( n )$ is
(A) $2 ^ { n } ( n + 1 )$
(B) $2 ^ { n } ( n + 1 ) - 1$
(C) $2 ^ { n } ( n - 1 )$
(D) $2 ^ { n } ( n - 1 ) + 1$.

Let $X$ be the set $\{ 1,2,3 , \ldots , 10 \}$ and $P$ the subset $\{ 1,2,3,4,5 \}$. The number of subsets $Q$ of $X$ such that $P \cap Q = \{ 3 \}$ is
(A) 1
(B) $2 ^ { 4 }$
(C) $2 ^ { 5 }$
(D) $2 ^ { 9 }$

Let $S = \{ 1,2,3,4 \}$. The total number of unordered pairs of disjoint subsets of $S$ is equal to
A) 25
B) 34
C) 42
D) 41

Let $S = \{ 1,2,3,4,5,6 \}$ and $X$ be the set of all relations $R$ from $S$ to $S$ that satisfy both the following properties: i. $R$ has exactly 6 elements. ii. For each $( a , b ) \in R$, we have $| a - b | \geq 2$. Let $Y = \{ R \in X$ : The range of $R$ has exactly one element $\}$ and $Z = \{ R \in X : R$ is a function from $S$ to $S \}$. Let $n ( A )$ denote the number of elements in a set $A$. If $n ( X ) = {}^{ m } C _ { 6 }$, then the value of $m$ is $\_\_\_\_$ .

Let $S = \{ 1,2,3,4,5,6 \}$ and $X$ be the set of all relations $R$ from $S$ to $S$ that satisfy both the following properties: i. $R$ has exactly 6 elements. ii. For each $( a , b ) \in R$, we have $| a - b | \geq 2$. Let $Y = \{ R \in X$ : The range of $R$ has exactly one element $\}$ and $Z = \{ R \in X : R$ is a function from $S$ to $S \}$. Let $n ( A )$ denote the number of elements in a set $A$. If the value of $n ( Y ) + n ( Z )$ is $k ^ { 2 }$, then $| k |$ is $\_\_\_\_$ .

Let $A$ and $B$ be two sets containing 2 elements and 4 elements respectively. The number of subsets of $A \times B$ having 3 or more elements is:
(1) 219
(2) 211
(3) 256
(4) 220

Let $A$ and $B$ be two sets containing four and two elements respectively. Then the number of subsets of the set $A \times B$, each having at least three elements is
(1) 510
(2) 219
(3) 256
(4) 275

Let $S = \{ 1,2,3 , \ldots , 100 \}$, then number of non-empty subsets $A$ of $S$ such that the product of elements in $A$ is even is :
(1) $2 ^ { 100 } - 1$
(2) $2 ^ { 50 } + 1$
(3) $2 ^ { 50 } \left( 2 ^ { 50 } - 1 \right)$
(4) $2 ^ { 50 } - 1$

Let $\cup _ { i = 1 } ^ { 50 } X _ { i } = \cup _ { i = 1 } ^ { n } Y _ { i } = T$, where each $X _ { i }$ contains 10 elements and each $Y _ { i }$ contains 5 elements. If each element of the set $T$ is an element of exactly 20 of sets $X _ { i }$'s and exactly 6 of sets $Y _ { i }$'s then $n$ is equal to:
(1) 15
(2) 50
(3) 45
(4) 30

Let $Z$ be the set of all integers, $A = \left\{ ( x , y ) \in Z \times Z : ( x - 2 ) ^ { 2 } + y ^ { 2 } \leq 4 \right\}$ $B = \left\{ ( x , y ) \in Z \times Z : x ^ { 2 } + y ^ { 2 } \leq 4 \right\}$ and $C = \left\{ ( x , y ) \in Z \times Z : ( x - 2 ) ^ { 2 } + ( y - 2 ) ^ { 2 } \leq 4 \right\}$ If the total number of relations from $A \cap B$ to $A \cap C$ is $2 ^ { p }$, then the value of $p$ is: (1) 25 (2) 9 (3) 16 (4) 49

Let the number of elements in sets $A$ and $B$ be five and two respectively. Then the number of subsets of $A \times B$ each having at least 3 and at most 6 elements is
(1) 752
(2) 782
(3) 792
(4) 772

Let $A$ and $B$ be two finite sets with $m$ and $n$ elements respectively. The total number of subsets of the set $A$ is 56 more than the total number of subsets of $B$. Then the distance of the point $\mathrm { P } ( \mathrm { m } , \mathrm { n } )$ from the point $\mathrm { Q } ( - 2 , - 3 )$ is
(1) 10
(2) 6
(3) 4
(4) 8

Q85. In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both Physics and Chemistry; 50 studied both Chemistry and Mathematics; 40 studied both Mathematics and Physics and 10 studied none of these subjects. Let m and n respectively be the least and the most number of students who studied all the three subjects. Then $\mathrm { m } + \mathrm { n }$ is equal to $\_\_\_\_$

$$\mathrm { X } \subseteq \{ \mathrm { a } , \mathrm {~b} , \mathrm { c } , \mathrm {~d} , \mathrm { e } \}$$
Given that, how many different subsets $X$ are there such that the number of elements in $\mathbf { X } \cap \{ \mathbf { a } , \mathbf { b } \}$ is 1?
A) 10 B) 12 C) 14 D) 16 E) 18

Let A be a subset of the set $\{ 1,2,3,4,5,6,7 \}$. $$A \cap \{ 5,6,7 \}$$ The elements of the set are odd numbers.\ Accordingly, how many three-element sets A satisfy this condition?\ A) 12\ B) 14\ C) 16\ D) 18\ E) 20