Let A = [1, 2, 3, 4] and R be a relation on the set A × A defined by R = { (a, b), (c, d) : 2a + 3b = 4c + 5d }. Then the number of elements in R is ______

To find the number of elements in the relation R defined on the set A = [1, 2, 3, 4], we need to analyze the equation given: 2a + 3b = 4c + 5d.

Understanding the Variables

Here, a, b, c, and d are elements from the set A. Since A has 4 elements, we can choose values for a, b, c, and d from {1, 2, 3, 4}.

Calculating Possible Combinations

We will evaluate all combinations of (a, b) and (c, d) to see how many satisfy the equation.

• For each pair (a, b), calculate 2a + 3b.
• For each pair (c, d), calculate 4c + 5d.
• Count how many pairs (a, b) equal the pairs (c, d) based on the equation.

Evaluating the Equation

Let's compute the values:

• For (1, 1): 2(1) + 3(1) = 5
• For (1, 2): 2(1) + 3(2) = 8
• For (1, 3): 2(1) + 3(3) = 11
• For (1, 4): 2(1) + 3(4) = 14
• For (2, 1): 2(2) + 3(1) = 7
• For (2, 2): 2(2) + 3(2) = 10
• For (2, 3): 2(2) + 3(3) = 13
• For (2, 4): 2(2) + 3(4) = 16
• For (3, 1): 2(3) + 3(1) = 9
• For (3, 2): 2(3) + 3(2) = 12
• For (3, 3): 2(3) + 3(3) = 15
• For (3, 4): 2(3) + 3(4) = 18
• For (4, 1): 2(4) + 3(1) = 11
• For (4, 2): 2(4) + 3(2) = 14
• For (4, 3): 2(4) + 3(3) = 17
• For (4, 4): 2(4) + 3(4) = 20

Finding Matches

Now, we need to find pairs (c, d) that yield the same results:

• 5: No pairs (c, d) yield this.
• 8: No pairs (c, d) yield this.
• 11: (1, 4) and (4, 1) yield this.
• 14: (1, 4) and (4, 2) yield this.
• 7: No pairs (c, d) yield this.
• 10: No pairs (c, d) yield this.
• 13: No pairs (c, d) yield this.
• 16: No pairs (c, d) yield this.
• 9: No pairs (c, d) yield this.
• 12: No pairs (c, d) yield this.
• 15: No pairs (c, d) yield this.
• 18: No pairs (c, d) yield this.
• 20: No pairs (c, d) yield this.

Final Count

After evaluating all combinations, we find that the valid pairs are:

• (1, 4) with (4, 1)
• (4, 1) with (1, 4)

Thus, the total number of elements in the relation R is 2.