- On Z+, define a ∗ b = ab
- On R – {– 1}, define a ∗ b = a/(b+1)
(v) On Z+, characterize a ∗ b = abdominal muscle
Stage 1: Check for commutative
Consider ∗ is commutative, then, at that point a ∗ b = b * a
Which implies, abdominal muscle = ba Which isn’t correct
a ∗ b = b * a for each of the a, b ∈ Z+
Consequently, ∗ isn’t commutative. Stage 2: Check for Associative.
Consider ∗ is affiliated, then, at that point (a ∗ b)* c = a * (b * c)
LHS = (abdominal muscle ) * c
= (abdominal muscle )c
RHS = a * (b * c) = a * (bc)
= ????????????
This suggests LHS ≠ RHS
Hence, ∗ isn’t cooperative.
(vi) On R – {–1}, characterize a ∗ b = a/(b+1)
Stage 1: Check for commutative
Consider ∗ is commutative, then, at that point a ∗ b = b * a
Which implies, a/(b+1) = b/(a+1) Which isn’t accurate
Consequently, ∗ is commutative.
Stage 2: Check for Associative.
Consider ∗ is affiliated, then, at that point (a ∗ b)* c = a * (b * c)
LHS = (a ∗ b) * c = (a/(b+1)) * c
????
= ????+1
????
= a/(c(b+1)
RHS = a * (b * c) = a * (b/(c + 1))
????
= ????
????+1
= a(c+1)/b
This suggests LHS ≠ RHS
Hence, ∗ isn’t cooperative parallel activity.