The question asks to compute a specific set-theoretic result (union, intersection, complement, difference, or Cartesian product) given explicitly defined sets.
Let $U = \{ 1,2,3,4,5,6 \} , A = \{ 1,3,6 \} , B = \{ 2,3,4 \}$. Then $A \cap \left( \complement_U B \right) =$ A. $\{ 3 \}$ B. $\{ 1,6 \}$ C. $\{ 5,6 \}$ D. $\{ 1,3 \}$
Let $A = \{ x \mid x = 3k + 1 , k \in Z \} , B = \{ x \mid x = 3k + 2 , k \in Z \} , U$ be the set of integers, then $C_{U}(A \bigcap B) =$ A. $\{ x \mid x = 3k , k \in Z \}$ B. $\{ x \mid x = 3k - 1 , k \in Z \}$ C. $\{ x \mid x = 3k - 1 , k \in \mathrm{Z} \}$ D. $\varnothing$
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 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. $8$
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) $\{ 4,5,7,8,9 \}$