MAT-3110 Midterm: MATH 3110 App State Fall2013 Midterm answer key
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Be sure to show your work! (a) let s = (0, 1] = {x | 0 < x 1}. Then s is not a group under multiplication. List the group axioms which hold and then using concrete counterexamples, show the other axioms fail. Since this works for all real numbers, it certainly holds for some real numbers (i. e. those in s): identity: we know that 1 is the multiplicative identity (so it acts as a multiplicative identity on s). Notice that in fact 1 s since 0 < 1 1. Axioms that fail: inverses counter-example: 0. 5 s but 0. 5 1 = 2 6 s since 2 > 1. Notice that given x s, x 1 = 1/x does in fact exist (as a real number). In other words, the inverse exists in some sense, but it does not necessarily belong to our set.
