Q: If $A = \{1, 2, 3\}$ and $B = \{3, 4\}$, then $(A \cup B) \times (A \cap B)$ is:

To find $(A \cup B) \times (A \cap B)$, first find the union and intersection of sets A and B. $A \cup B = \{1, 2, 3, 4\}$ and $A \cap B = \{3\}$. Then the Cartesian product is $\{1, 2, 3, 4\} \times \{3\} = \{(1, 3), (2, 3), (3, 3), (4, 3)\}$. Therefore, the correct option is C.

Q: If $P = \{1, 2\}$, form the set $P \times P \times P$.

The Cartesian product of three sets $P \times P \times P$ consists of all ordered triplets $(a, b, c)$ where $a, b, c \in P$. Since $P = \{1, 2\}$, the number of elements in $P \times P \times P$ is $2 \times 2 \times 2 = 8$. Listing them out gives $\{(1,1,1), (1,1,2), (1,2,1), (1,2,2), (2,1,1), (2,

Q: Let $A = \{1, 2\}$ and $B = \{3, 4\}$. The number of relations from $A$ to $B$ is:

The number of elements in $A \times B$ is $n(A) \times n(B) = 2 \times 2 = 4$. The total number of relations from $A$ to $B$ is equal to the number of subsets of $A \times B$, which is $2^{n(A \times B)} = 2^4 = 16$. Thus, option D is correct.

Question 1

If $A = \{1, 2, 3\}$ and $B = \{3, 4\}$, then $(A \cup B) 	imes (A \cap B)$ is:

Accepted Answer

To find $(A \cup B) 	imes (A \cap B)$, first find the union and intersection of sets A and B. $A \cup B = \{1, 2, 3, 4\}$ and $A \cap B = \{3\}$. Then the Cartesian product is $\{1, 2, 3, 4\} 	imes \{3\} = \{(1, 3), (2, 3), (3, 3), (4, 3)\}$. Therefore, the correct option is C.

Question 2

If $P = \{1, 2\}$, form the set $P 	imes P 	imes P$.

Accepted Answer

The Cartesian product of three sets $P 	imes P 	imes P$ consists of all ordered triplets $(a, b, c)$ where $a, b, c \in P$. Since $P = \{1, 2\}$, the number of elements in $P 	imes P 	imes P$ is $2 	imes 2 	imes 2 = 8$. Listing them out gives $\{(1,1,1), (1,1,2), (1,2,1), (1,2,2), (2,1,1), (2,

Question 3

If $A 	imes B = \{(a, x), (a, y), (b, x), (b, y)\}$, find $A$ and $B$.

Accepted Answer

The set $A$ consists of all first elements of the ordered pairs in $A 	imes B$, so $A = \{a, b\}$. The set $B$ consists of all second elements of the ordered pairs in $A 	imes B$, so $B = \{x, y\}$. Therefore, option A is correct.

Question 4

Let $A = \{1, 2\}$ and $B = \{3, 4\}$. The number of relations from $A$ to $B$ is:

Accepted Answer

The number of elements in $A 	imes B$ is $n(A) 	imes n(B) = 2 	imes 2 = 4$. The total number of relations from $A$ to $B$ is equal to the number of subsets of $A 	imes B$, which is $2^{n(A 	imes B)} = 2^4 = 16$. Thus, option D is correct.

Question 5

Let $R$ be the relation on $\mathbb{N}$ defined by $R = \{(x, y) : y = x + 5, x

Accepted Answer

The condition given is $x

Relations and Functions