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Step 1: Understand the Equivalence Relation
The given set is A = {2, 3, 4, 5, ā¦, 30}. The relation '$\simeq$' on AĆA is defined by (a, b) $\simeq$ (c, d) if and only if $ad = bc$. We need to find how many ordered pairs (a, b) in AĆA are equivalent to (4, 3) under this relation.
Step 2: Apply the Condition to (4, 3)
For (a, b) to be equivalent to (4, 3), we have:
$a \cdot 3 = b \cdot 4$
which simplifies to
$3a = 4b$
or
$a = \frac{4}{3} \times b$.
Step 3: Determine Valid Values of b
Since a and b must both lie in the set A = {2, 3, 4, 5, ā¦, 30}, and $a = \frac{4}{3}b$, b must be a multiple of 3 for a to be an integer. Thus we check multiples of 3 in A:
b can be 3, 6, 9, 12, 15, 18, 21, 24, 27, 30.
Step 4: Calculate Corresponding Values of a and Verify They Are in A
For b = 3, $a = \frac{4}{3} \times 3 = 4$ (valid, 4 ā A)
For b = 6, $a = \frac{4}{3} \times 6 = 8$ (valid, 8 ā A)
For b = 9, $a = \frac{4}{3} \times 9 = 12$ (valid, 12 ā A)
For b = 12, $a = \frac{4}{3} \times 12 = 16$ (valid, 16 ā A)
For b = 15, $a = \frac{4}{3} \times 15 = 20$ (valid, 20 ā A)
For b = 18, $a = \frac{4}{3} \times 18 = 24$ (valid, 24 ā A)
For b = 21, $a = \frac{4}{3} \times 21 = 28$ (valid, 28 ā A)
For b = 24, $a = \frac{4}{3} \times 24 = 32$ (not valid, 32 ā A)
For b = 27, $a = \frac{4}{3} \times 27 = 36$ (not valid, 36 ā A)
For b = 30, $a = \frac{4}{3} \times 30 = 40$ (not valid, 40 ā A)
Step 5: List All Valid Ordered Pairs
The valid pairs (a, b) that satisfy (a, b) $\simeq$ (4, 3) are:
(4, 3), (8, 6), (12, 9), (16, 12), (20, 15), (24, 18), (28, 21).
There are 7 such ordered pairs.
Step 6: Conclusion
The number of ordered pairs (a, b) in AĆA that are equivalent to (4, 3) under the relation (ad = bc) is 7.
