For each of the following sets and proposed binary operation, determine whether or not this defines a group. Provide either proofs or an explanation of why this does not define a group. [You should not assume that the proposed binary operation is indeed a binary operation. Verifying that it is indeed a binary operation would be one step in a proof.] The set {x ER | −1 < x < 1, x # 0} under multiplication of real numbers. The set {x EZ | x = 1 (mod 3) } under addition. The set \left\{\left(\begin{array}{ll} a & 0 \\ 0 & 0 \end{array}\right) \mid a \in \mathbb{R}, a \neq 0\right\} under matrix multiplication. The set (X) of all subsets of a non-empty set X under the operation of intersection. An arbitrary set A with |A > 1 under the operation * defined by xy = x for allx, y = A.

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