1. Which of the following maps define an isomorphism of binarystructures? Explain. (a) The function (R, +) ? (R, +) given by x ?x 2 (b) The function (R>0, ·) ? (R>0, ·) given by x ? x 2 (c)The function (R>0, +) ? (R>0, +) given by x ? 2x 2. For eachof the following, prove or disprove that it is a group. If it is agroup, prove or disprove that it is abelian. (a) (Q +, ·) (b) (R {0}, ?), where a ? b = ab 2 3. Assume that ? : (S, ?) ? (S 0 , ? 0) is an isomorphism of binary structures, and that ? isassociative. Prove that ? 0 is associative. 4. Consider the set H =a ?b b a ? M2(R) : a 6= 0 or b 6= 0 , with matrix multiplication asthe operation. (a) Show that the operation is closed. (b) Definethe function ? : H ? C ? (where C ? is the group C {0} withmultiplication as the operation) by ? : a ?b b a 7? a + bi. Showthat ? is an isomorphism of binary structures. (c) Explain brieflywhy we can now conclude that H is a group. 5. Consider the set V =a 0 0 b ? M2(R) : a, b ? {1, ?1} , with the operation · (matrixmultiplication). Note that (V, ·) is a group (you do not need toprove this). Prove that (V, ·) is not isomorphic to U4.
