It is given that *: R * R -> R and o: R * R -> R is defined as

a * b = |a - b| and a o b = a for all a, b ∈ R

Now, for a, b ∈ R, we have

a * b = |a - b|

and b * a = |b - a| = |-(a - b)| = |a - b| 

Since, a * b = b * a

Hence, operation * is commutative.

Again, (1 * 2) * 3 = |(1 - 2)| * 3 = |-1| * 3 = 1 * 3 = 3    {1, 2, 3 ∈ R}

and 1 * (2 * 3) = 1 * |(2 - 3)| = 1 * |-1| = 1 * 1 = |1 - 1| = 0

Since, (1 * 2) * 3 ≠ 1 * (2 * 3)

So, the operation * is not associative

Now, 1 o 2 = 1   {1, 2 ∈ R}

and 2 o 1 = 2

Since, 1 o 2 ≠ 2 o 1 

So, the operation o is not commutative.

Again, let a, b, c ∈ R, we have

(a o b) o c = a o c = a

a o (b o c) = a o b = a

Since, (a o b) o c = a o (b o c) = a

So, the operation o is associative

Now, let a, b, c ∈ R, we have

a * (b o c) = a * b = |a - b|

(a * b) o (a o c) = (|a - b|) o (|a - c|) = |a - b|

Hence, a * (b o c) = (a * b) o (a o c)

Again, 1 o (2 * 3) = 1 o (|2 - 3|) = 1 o |-1| = 1 o 1 = 1    {1, 2, 3 ∈ R}

(1 o 2) * (1 o 3) = 1 * 1 = |1- 1| = 0

Since, 1 o (2 * 3) ≠ (1 o 2) * (1 o 3) 

So, the operatoion o is not distribute over *