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4. Let S a b 1 1 (a) List the elements of Z. (b) List the elements of S. (c) Make a Cayley table for (S. where X is matrix multiplication and the addition and multiplication of matrix entries within the matrix multiplication is done modulo 3 (so that the results will still be elements of (d) Is (S, x) commutative? (e) Does (S, x) have an identity? If so, what is it? (f) Which elements of (S, x) are units? Find the inverse of each unit. (g) Is (S, x) a group? 5. For each of the following pairs (M, *D consisting of a set M and a operation * on M, determine whether (M, *) is a monoid, a group, or neither. (a) (21 x) (b) (P(U), U), where U is a set and POUD is the power set of U (c) (R. *), where * is defined by . * y . + y - my of 7

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