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:
We are asked to find the total number of mathematical relations from set \( A \) to set \( B \).


Step 2: Key Formula or Approach:
If a set \( A \) has \( m \) elements and a set \( B \) has \( n \) elements, then the number of elements in their Cartesian product \( A \times B \) is \( m \times n \).
A relation from \( A \) to \( B \) is defined as any subset of the Cartesian product \( A \times B \).
The total number of relations is simply the number of all possible subsets of \( A \times B \), which evaluates to \( 2^{m \times n} \).


Step 3: Detailed Explanation:
First, let's determine the specific elements of set \( A \).
\( A = \{ \text{even natural numbers less than 8} \} \)
\( A = \{2, 4, 6\} \)
So, the number of elements in \( A \), denoted as \( |A| \), is 3.
Next, let's determine the elements of set \( B \).
\( B = \{ \text{prime numbers less than 7} \} \)
\( B = \{2, 3, 5\} \)
So, the number of elements in \( B \), denoted as \( |B| \), is 3.
The total number of possible relations from \( A \) to \( B \) is given by:
\[ 2^{|A| \times |B|} = 2^{3 \times 3} = 2^9 \]

Step 4: Final Answer:
The number of relations from \( A \) to \( B \) is \( 2^9 \).

Answer 2

A relation from set \( A \) to set \( B \) is any collection of ordered pairs \( (a, b) \) with \( a \in A \) and \( b \in B \). We can count these directly by listing every possible pair and treating the choice of whether to include each one as an independent yes-or-no decision.

First, list the elements: \( A = \{2, 4, 6\} \) and \( B = \{2, 3, 5\} \). Every ordered pair \( (a, b) \) with \( a \in A \), \( b \in B \) is:

\[ (2,2), (2,3), (2,5), (4,2), (4,3), (4,5), (6,2), (6,3), (6,5) \]

This is a list of exactly \( 9 \) distinct ordered pairs. Now, to build a relation, we go through this list one pair at a time and decide, independently, whether that pair belongs to the relation or not, a simple yes-or-no choice for each of the \( 9 \) pairs.

Since there are \( 2 \) choices (include or exclude) for each of the \( 9 \) pairs, and these choices are made independently of one another, the total number of distinct relations is:

\[ 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^9 \]

This confirms that the total number of relations from \( A \) to \( B \) is \( 2^9 \).

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