CBSE Class 12 Mathematics Relations And Functions Class Test Set 15

Question. Consider the binary operation \( * \) on the set \( \{1,2,3,4,5\} \) defined by \( a * b = \min \{a, b\} \). Write the operational table.
Answer: The set is \( A = \{1, 2, 3, 4, 5\} \) and the operation is defined by \( a * b = \min \{a, b\} \).
The operational table for the given binary operation is as follows:

Question. If the binary operation \( * \) on the set of integers \( \mathbb{Z} \), is defined by \( a * b = a + 3b^2 \), then find the value of \( 2 * 4 \).
Answer: Given binary operation is:
\( a * b = a + 3b^2 \)

To find the value of \( 2 * 4 \), we substitute \( a = 2 \) and \( b = 4 \):
\( 2 * 4 = 2 + 3(4)^2 \)
\( 2 * 4 = 2 + 3(16) \)
\( 2 * 4 = 2 + 48 \)
\( 2 * 4 = 50 \)

Question. Determine which of the following binary operations on the set \( \mathbb{N} \) are associative and which are commutative.
(i) \( a * b = 1 \quad \forall\ a, b \in \mathbb{N} \)
(ii) \( a * b = \frac{a+b}{2} \quad \forall\ a, b \in \mathbb{N} \)
Answer:
(i) \( a * b = 1 \quad \forall\ a, b \in \mathbb{N} \)

• Commutativity:
  For any \( a, b \in \mathbb{N} \):
  \( a * b = 1 \)
  \( b * a = 1 \)
  Since \( a * b = b * a \), the operation is commutative.
• Associativity:
  For any \( a, b, c \in \mathbb{N} \):
  \( (a * b) * c = 1 * c = 1 \)
  \( a * (b * c) = a * 1 = 1 \)
  Since \( (a * b) * c = a * (b * c) \), the operation is associative.

Hence, operation (i) is both commutative and associative.

(ii) \( a * b = \frac{a+b}{2} \quad \forall\ a, b \in \mathbb{N} \)

• Commutativity:
  For any \( a, b \in \mathbb{N} \):
  \( a * b = \frac{a+b}{2} \)
  \( b * a = \frac{b+a}{2} = \frac{a+b}{2} \)
  Since \( a * b = b * a \), the operation is commutative.
• Associativity:
  For any \( a, b, c \in \mathbb{N} \):
  \( (a * b) * c = \left(\frac{a+b}{2}\right) * c = \frac{\frac{a+b}{2} + c}{2} = \frac{a+b+2c}{4} \)
  \( a * (b * c) = a * \left(\frac{b+c}{2}\right) = \frac{a + \frac{b+c}{2}}{2} = \frac{2a+b+c}{4} \)
  Let us check with an example: Let \( a = 1, b = 2, c = 3 \).
  \( (1 * 2) * 3 = \frac{1+2+2(3)}{4} = \frac{9}{4} \)
  \( 1 * (2 * 3) = \frac{2(1)+2+3}{4} = \frac{7}{4} \)
  Since \( (a * b) * c \neq a * (b * c) \), the operation is not associative.

Hence, operation (ii) is commutative but not associative.

Question. If \( a * b = a + ab \), find \( 1 * (2 * 3) \).
Answer: Given operation is \( a * b = a + ab \).
First, we find the value of \( 2 * 3 \):
\( 2 * 3 = 2 + 2(3) \)
\( 2 * 3 = 2 + 6 = 8 \)

Now, we find \( 1 * (2 * 3) \), which is \( 1 * 8 \):
\( 1 * 8 = 1 + 1(8) \)
\( 1 * 8 = 1 + 8 = 9 \)
Therefore, \( 1 * (2 * 3) = 9 \).