Question

Let \(A\) be the set of odd integers in \([0,10]\) and let \(B\) be the set of prime numbers in \([0,10]\). If \[ R=\{(a,b)\in A\times B:\ a+b \text{ is odd}\}, \] then \(n(R)\) is equal to:

• A. 1
• B. 2
• C. 3
• D. 4
• E. 5

Hint:

In competitive exams, remember that \(2\) is the only even prime number. If a condition requires a prime number to be even to satisfy a parity rule, it immediately locks \(b = 2\), simplifying the counting process significantly.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
First, we need to explicitly list the elements of set A (odd integers) and set B (prime numbers) within the given interval \([0, 10]\).
A relation R is defined based on a condition: the sum of the elements in the pair \((a, b)\) must be odd.

Step 2: Key Formula or Approach:
Rule of parity for addition:
Sum is Odd \(\iff\) (Odd + Even) or (Even + Odd).
Since set A consists strictly of odd integers, for \(a + b\) to be odd, \(b\) must be an even integer from set B.

Step 3: Detailed Explanation:
Listing elements of A:
Odd integers in \([0, 10]\) are \(A = \{1, 3, 5, 7, 9\}\).
Listing elements of B:
Prime numbers in \([0, 10]\) are \(B = \{2, 3, 5, 7\}\).
Filtering elements for relation R:
Since every \(a \in A\) is odd, we need \(b \in B\) to be even.
In set B, the only even prime number is \(2\).
So, the pairs belonging to R are:
\[ R = \{ (1, 2), (3, 2), (5, 2), (7, 2), (9, 2) \} \]
Counting the elements:
The number of elements in R is \(n(R) = 5\).

Step 4: Final Answer:
The value of \(n(R)\) is 5.