Question

Determine whether the binary operation ∗ on R defined by a ∗ b = |a – b| is commutative. Also, find the value of (–3) ∗ 2.

Answer 1

To determine whether the binary operation _* on R defined by a ∗ b = |a – b| is commutative, we need to check if a ∗ b = b ∗ a for all real numbers a and b.

Step 1: The operation is defined as:

a ∗ b = |a – b|

Step 2: To check if the operation is commutative, we need to evaluate both a ∗ b and b ∗ a:

a ∗ b = |a – b|

b ∗ a = |b – a|

Step 3: Use properties of absolute values.

We know that:

|b – a|

= |– (a – b)| 

= |a – b|

This shows that:

b ∗ a

= |b – a|

= |a – b|

= a ∗ b

Since a ∗ b = b ∗ a for all a and b, we conclude that the operation is commutative.

Step 4: Calculate the value of (–3) ∗ 2. Now, we need to find the value of (–3) ∗ 2:

(–3) ∗ 2

= |–3 – 2|

= |–5|

= 5

Thus, the value of (–3) ∗ 2 is 5.