Let · be the binary operation on the set S = {a, b, c} de ned by the multipli-cation table below:· a b ca b c cb a c cc c c cFor each s ∈ S de ne an operation ∗s by x ∗s y = x · (s · y) and de ne anoperation ◦s by x ◦s y = (x · s) · y. The multiplication tables for ∗a, ◦a arepartially completed below.∗a a b ca c cb c c cc c c◦a a b ca c ab c cc c c c(i) Complete the missing entries. [4 marks](ii) Can Light's Associativity test, together with the completed tables for∗a, ◦a, be used to determine whether or not the operation · is associative?If `yes', say if the operation is associative, justifying your answer; if `no',explain why not.
Question
Let · be the binary operation on the set S = {a, b, c} de ned by the multipli-cation table below:· a b ca b c cb a c cc c c cFor each s ∈ S de ne an operation ∗s by x ∗s y = x · (s · y) and de ne anoperation ◦s by x ◦s y = (x · s) · y. The multiplication tables for ∗a, ◦a arepartially completed below.∗a a b ca c cb c c cc c c◦a a b ca c ab c cc c c c(i) Complete the missing entries. 4 marks Can Light's Associativity test, together with the completed tables for∗a, ◦a, be used to determine whether or not the operation · is associative?If yes', say if the operation is associative, justifying your answer; if no',explain why not.
Solution
The question seems to be cut off and incomplete. However, I'll try to answer based on the information given.
(i) To complete the missing entries, we need to use the definition of the operations ∗a and ◦a and the given multiplication table for the operation ·.
For ∗a, we have x ∗a y = x · (a · y). So, for example, b ∗a a = b · (a · a) = b · a = c. Similarly, c ∗a a = c · (a · a) = c · a = c. We can fill in the rest of the table in the same way.
For ◦a, we have x ◦a y = (x · a) · y. So, for example, b ◦a a = (b · a) · a = c · a = c. Similarly, c ◦a a = (c · a) · a = c · a = c. We can fill in the rest of the table in the same way.
(ii) Light's Associativity test is a method for checking if a binary operation is associative. However, it requires that we know the operation for all pairs of elements in the set. In this case, we only know the operation for pairs involving a. Therefore, we cannot use Light's Associativity test to determine whether or not the operation · is associative.
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