Question

Let $A=\{2,3,4,5\}$, $B=\{36,45,49,60,77,90\}$ and let $R$ be the relation 'is factor of' from $A$ to $B$. Then the range of $R$ is the set

• A. {60}
• B. {36,45,60,90}
• C. {49,77}
• D. {49,60,77}
• E. {36,45,49,60,77,90}

Hint:

Logic Tip: To find the range quickly without writing all pairs, just look at each number in set B and ask, "Does this number have at least one factor in set A?"

Answer 1

The Correct Option is B

Concept:
A relation $R$ from $A$ to $B$ is a set of ordered pairs $(a, b)$ where $a \in A$ and $b \in B$. The range of $R$ is the set of all second elements $b$ that have at least one valid mapping from $A$ according to the rule "$a$ is a factor of $b$".

Step 1: Check factors for a = 2.
Test which elements in B are divisible by 2 (even numbers): 36, 60, and 90 are divisible by 2. Pairs generated: $(2, 36), (2, 60), (2, 90)$

Step 2: Check factors for a = 3.
Test which elements in B are divisible by 3 (sum of digits is multiple of 3): 36, 45, 60, and 90 are divisible by 3. Pairs generated: $(3, 36), (3, 45), (3, 60), (3, 90)$

Step 3: Check factors for a = 4 and 5.
Test divisibility for 4 and 5: For 4: 36 and 60. Pairs: $(4, 36), (4, 60)$ For 5: 45, 60, and 90. Pairs: $(5, 45), (5, 60), (5, 90)$

Step 4: Compile the complete relation R.
Combine all pairs to form the relation set $R$: $R = \{(2,36), (2,60), (2,90), (3,36), (3,45), (3,60), (3,90), (4,36), (4,60), (5,45), (5,60), (5,90)\}$

Step 5: Extract the range.
The range consists of all unique values appearing as the second coordinate in the pairs of $R$: $$\text{Range} = \{36, 45, 60, 90\}$$ Notice 49 and 77 are excluded because they have no factors in set A. Hence the correct answer is (B) {36,45,60,90\.