If and , then find the number of elements in .

30 Explanation: Solution To find the number of elements in the Cartesian product of multiple sets, we use the formula: Step 1: Find (Union of sets) The union contains all unique elements present in either or . Step 2: Find (Intersection of sets) The intersection contains only the elements common to both and . Step 3: Find (Symmetric Difference) The symmetric difference is defined as or . It contains elements that are in or , but not in both. Step 4: Calculate the total number of elements. Now, we multiply the cardinalities (number of elements) of each set: Final Answer: The number of elements is .

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NIMCET 2021 — Mathematics PYQ

NIMCET | Mathematics | 2021

If $A = {1, 2, 3, 4}$ and $B = {3, 4, 5}$ , then find the number of elements in $(A \cup B) \times (A \cap B) \times (A Δ B)$ .

Choose the correct answer:

Explanation

Solution

To find the number of elements in the Cartesian product of multiple sets, we use the formula:

n(X \times Y \times Z) = n(X) \times n(Y) \times n(Z)

Step 1: Find $A \cup B$ (Union of sets)

The union contains all unique elements present in either $A$ or $B$ .

A \cup B = \{1, 2, 3, 4, 5\}

n(A \cup B) = 5

Step 2: Find $A \cap B$ (Intersection of sets)

The intersection contains only the elements common to both $A$ and $B$ .

A \cap B = \{3, 4\}

n(A \cap B) = 2

Step 3: Find $A \Delta B$ (Symmetric Difference)

The symmetric difference $A \Delta B$ is defined as $(A - B) \cup (B - A)$ or $(A \cup B) - (A \cap B)$ . It contains elements that are in $A$ or $B$ , but not in both.

A \Delta B = \{1, 2, 3, 4, 5\} - \{3, 4\}

A \Delta B = \{1, 2, 5\}

n(A \Delta B) = 3

Step 4: Calculate the total number of elements.

Now, we multiply the cardinalities (number of elements) of each set:

n((A \cup B) \times (A \cap B) \times (A \Delta B)) = n(A \cup B) \times n(A \cap B) \times n(A \Delta B)

= 5 \times 2 \times 3

= 10 \times 3

= 30

Final Answer:

The number of elements is $30$ .