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 $2^9$.