In each ease, state whether or not the given set and binary operation form a gro
ID: 3009715 • Letter: I
Question
In each ease, state whether or not the given set and binary operation form a group. If so, determine the identity element e. If not, identify a group property that fails and explain why. R^+ under addition. The set 3Z = {x | x = 3k for some k epsilon Z} under addition. R - {0} under the operation a * b = |ab|. The set {1, -1} under multiplication. The set R times R under the operation (x, y) * (z, w) = (x + z, y - w). The set R times (R - {0}) under the operation (x, y) * (z, w) = (x + z, yw). R - {1} under the operation a * b = a + b - ab. Z under the operation a * b = a + b - 1. The power set P(X) a non-empty set X under the operation A * B = A intersection B.Explanation / Answer
for group we need to check if given set follows four properties ( closure, associative, identity, inverse) or not.
Ans(a):
No R+ doesn't form group under addition because it fails inverse property as to get identity element 0, we need negative numbers while there is no negative number in R+
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Ans(b):
given set will have elements like {...,-6,-3,0,3,6,...}
Yes It forms group under addition and identity element is 0.
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Ans(c):
yes It forms group under given operation and identity element is 1.
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Ans(d):
No {1,-1} doesn't form group under multiplication because it fails inverse property as to get identity element 1,
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