Question

Let A = {1,2,3,4,5} and let B={1,2,3,4}. If the relation R : A → B is given by (a, b) ∈ R if and only if a+b is even, then n(R) is equal to

• A. 10
• B. 16
• C. 20
• D. 12
• E. 6

Answer 1

The Correct Option is A

We are given two sets:

\( A = \{1, 2, 3, 4, 5\} \) and \( B = \{1, 2, 3, 4\} \)

The relation \( R \) is defined by the condition that \( (a, b) \in R \) if and only if \( a + b \) is even.

Now, let's consider all possible pairs where \( a + b \) is even:

For \( a = 1 \):

• 1 + 1 = 2 (even) => valid pair (1, 1)
• 1 + 3 = 4 (even) => valid pair (1, 3)

For \( a = 2 \):

• 2 + 2 = 4 (even) => valid pair (2, 2)
• 2 + 4 = 6 (even) => valid pair (2, 4)

For \( a = 3 \):

• 3 + 1 = 4 (even) => valid pair (3, 1)
• 3 + 3 = 6 (even) => valid pair (3, 3)

For \( a = 4 \):

• 4 + 2 = 6 (even) => valid pair (4, 2)
• 4 + 4 = 8 (even) => valid pair (4, 4)

For \( a = 5 \):

• 5 + 1 = 6 (even) => valid pair (5, 1)
• 5 + 3 = 8 (even) => valid pair (5, 3)

Thus, the total number of valid pairs is \( 2 + 2 + 2 + 2 + 2 = 10 \).

The correct option is (A) : 10

Answer 2

We are given two sets: A = {1, 2, 3, 4, 5} and B = {1, 2, 3, 4}. The relation R: A → B is defined such that (a, b) ∈ R if and only if a + b is even.

For a + b to be even, both a and b must be either even or odd. Let's identify the even and odd numbers in A and B:

• A (Odd): {1, 3, 5} - There are 3 odd numbers.
• A (Even): {2, 4} - There are 2 even numbers.
• B (Odd): {1, 3} - There are 2 odd numbers.
• B (Even): {2, 4} - There are 2 even numbers.

Now we create pairs (a, b) that satisfy the condition a + b is even:

• Odd + Odd: We can pair each of the 3 odd numbers in A with each of the 2 odd numbers in B. This gives us 3 * 2 = 6 pairs.
• Even + Even: We can pair each of the 2 even numbers in A with each of the 2 even numbers in B. This gives us 2 * 2 = 4 pairs.

The total number of ordered pairs in the relation R, n(R), is the sum of these pairs:

n(R) = 6 + 4 = 10

Therefore, n(R) is equal to 10.