c) and e) please (i) State what it means for an element a e 5 to have ar oers [5

Question

c) and e) please

(i) State what it means for an element a e 5 to have ar oers [5 marks c)Let S = {a,b,c,d), and define a binary operation * on S by the multiplication table la bc d a d ab b b a b cd a c d b cl d b d a c (i) Find an identity element of S with respect to (ii) Determine which (if any) elements have an inverse, in each case giving the inverse. (ii) Determine whether is commutative, briefly justifying your answer. (iv) Show that * is not associative. 17 marks] (d) Explain what is meant by "(G, *) is a group". 14 marks (e) Let G= {a, b, c} and let * be a binary operation on G. Comiplete the following multiplication table so that (G, ) forms a group: * a b c a la b c 3 marks

Explanation / Answer

C).

1)the identity element is b as in the given binary operation, if we multiply b with a, b, c, d gives a, b, c, d.

2)the possible inverse of a are c and b. But when inverse exists then it is unique. Hence inverse of a doesn't exit. The inverse of c is d;inverse of d is a.

3) No. This is not commutative because a*c=b, c*a=a, that the elements a and c are not commute.

4).now (a*c) *d=b*d=d but a*(c*d) =a*b=a. Hence they are not equal. So * is not associative.

E) since a group of order 3 is always cyclic and by the given data we easily see that a is the identity element.

Then the second row is : b c a

The third row is. :c a b

Then the following table (G, *) forms a group.

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