MAT-3110 Midterm: MATH 3110 App State Fall2015 Test1 answer key
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Name: answer key: (20 points) de nition and basics. Be sure to show your work! (a) suppose that g is a non-empty set equipped an operation. 1: closure: x, y g, we have xy g. 2: associativity: x, y, z g, we have (xy)z = x(yz) 3: identity: e g such that x g, xe = x = ex. 4: inverses: x g, y g such that xy = e = yx. 5: commutivity: x, y g, xy = yx (b) let g = z>0 be the set of positive integers. De ne a binary operation as follows: x y = xy (example: 2 3 = 23 = 8). G paired with this operation is not a group. The operation is closed: a positive integer power of a positive integer is a positive integer so no help here. To show g isn"t a group, the simplest answer is: is not associative .