Exercise: 5-D
Assertion-Reason Questions
Q1: Assertion (A): If A = {x : x is a letter of the word GEOLOGY} and B = { x : x is a factor of 16}, then A and B are equivalent.
Reason (R): The intersection of two equivalent sets is an empty set.
Step 1: Find set \(A\):
\(A = \{G, E, O, L, O, G, Y\} = \{G, E, O, L, Y\}\) (unique letters only)
Number of elements, \(n(A) = 5\)
Step 2: Find set \(B\):
Factors of 16 are \(1, 2, 4, 8, 16\), so
\(B = \{1, 2, 4, 8, 16\}\)
Number of elements, \(n(B) = 5\)
Step 3: Are \(A\) and \(B\) equivalent?
Two sets are equivalent if they have the same number of elements.
Here, \(n(A) = n(B) = 5\), so \(A\) and \(B\) are equivalent sets.
Step 4: Check Reason (R):
The intersection of two equivalent sets need not be empty.
For example, if two sets are equal, their intersection is the same as the set itself.
Thus, Reason (R) is false.
Answer: c. Assertion (A) is true but Reason (R) is false.
Q2: Assertion (A): The complement of an empty set does not exist.
Reason (R): The intersection of a set and its complement is a null set.
Step 1: The complement of a set \(A\), denoted \(A’\), is defined relative to a universal set \(\xi\) as all elements in \(\xi\) not in \(A\).
Step 2: The complement of the empty set \(\phi\) is the universal set \(\xi\) itself, because no elements are in \(\phi\), so all elements of \(\xi\) are outside it.
Therefore, the complement of the empty set does exist and equals \(\xi\).
Step 3: The intersection of a set \(A\) and its complement \(A’\) is always \(\phi\) (null set), since they have no elements in common.
Answer: d. Assertion (A) is false but Reason (R) is true.
Q3: Assertion (A): The set of natural a sub set of the set of whole numbers (W).
Reason (R): 0 is the only whole number which is not a natural number.
Step 1: Natural numbers are usually defined as \(N = \{1, 2, 3, \ldots\}\).
Whole numbers are defined as \(W = \{0, 1, 2, 3, \ldots\}\).
Step 2: Since every natural number is a whole number except 0, which is in \(W\) but not in \(N\), it follows that \(N \subset W\).
Step 3: Therefore, both Assertion and Reason are true, and Reason correctly explains Assertion.
Answer: a. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
