Question

Let \( A \) be the set of even natural numbers less than 8 and \( B \) be the set of prime numbers less than 7. The number of relations from \( A \) to \( B \) is

• A. \( 2^9 \)
• B. \( 2^9 - 1 \)
• C. \( 9^2 \)
• D. \( 9^2 - 1 \)

Hint:

The total number of relations from a set with \( m \) elements to a set with \( n \) elements is exactly the size of the power set of their Cartesian product, which is always \( 2^{mn} \).

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The topic here is Relations and Functions within Set Theory. The question asks for the total number of possible relations that can be formed between two specific sets, A and B. A "relation" from set A to set B is formally defined as any subset of the Cartesian product $A \times B$. Therefore, the total number of relations is equivalent to the total number of subsets of $A \times B$. To solve this, we must first determine the elements of each set and then calculate the size of their product.
Step 2: Key Formulas and approach:
The following formulas are essential:
1. If $n(A) = p$ and $n(B) = q$, then the number of elements in the Cartesian product $n(A \times B) = p \times q$.
2. The number of subsets of a set containing 'n' elements is $2^n$.
3. Therefore, the total number of relations from A to B is $2^{n(A) \times n(B)}$.
The approach involves listing the elements of A and B, counting them, and then applying the exponential formula.
Step 3: Detailed Explanation:

Define set A: Even natural numbers less than 8. These are $\{2, 4, 6\}$. So, the number of elements $n(A) = 3$.

Define set B: Prime numbers less than 7. These are $\{2, 3, 5\}$. Note that 1 is not prime and 7 is not "less than" 7. So, $n(B) = 3$.

Calculate the total number of ordered pairs in the Cartesian product $A \times B$. This is $3 \times 3 = 9$.

Since every unique subset of this Cartesian product represents a distinct relation, we use the power set formula.

Total relations = $2^{n(A \times B)} = 2^{3 \times 3} = 2^9$.

Step 4: Final Answer:
The total number of relations from set A to set B is $2^9$.

Answer 2

Since \( A = \{2, 4, 6\} \) and \( B = \{2, 3, 5\} \), the Cartesian product \( A \times B \) has \( 3 \times 3 = 9 \) ordered pairs, and every relation from \( A \) to \( B \) is one of the subsets of this \( 9 \)-element set. Let's check each option against this fact.

1. \( 2^9 \): This is exactly the number of subsets of a \( 9 \)-element set, which correctly counts every possible relation, including the empty relation and the full relation \( A \times B \) itself.
2. \( 2^9 - 1 \): This would be the count if we excluded one particular subset, such as the empty relation. But the empty set is a perfectly valid relation, since it is still a subset of \( A \times B \), so it should not be excluded. This undercounts the true total.
3. \( 9^2 \): This equals \( 81 \), which comes from squaring the number of pairs in \( A \times B \) rather than counting how many subsets that set of pairs has. Squaring the pair count has no bearing on the subset-counting definition of a relation, so this number does not represent a valid count here.
4. \( 9^2 - 1 \): This is simply one less than option (C) and inherits the same flaw: it is not derived from counting subsets of the \( 9 \) pairs, so it cannot be the correct count of relations either.

Only \( 2^9 \) correctly represents the number of subsets of the \( 9 \)-element Cartesian product \( A \times B \), which is precisely the number of relations from \( A \) to \( B \).

Therefore, the correct answer is \( 2^9 \).