Answer to Question #238989 in Discrete Mathematics for Joseph

Question #238989

Determine whether the following relations are injective and/or subjective function. Find universe of the  functions if they exist.

i. A= v,w,x,y,z, B=1,2,3,4,5

R= (v,z),(w,1), (x,3),(y,5)

ii. A = 1,2,3,4,5 B=1,2,3,4,5

R = (1,2),(2,3),(3,4),(4,5),(5,1)


1
Expert's answer
2021-09-20T16:48:47-0400

(I) Since R is mapping A to B. Then R must be of the form;


"R=\\{(a,b): a\\in A, b\\in B\\}"


But, the first element of R, (v,z) does not obeya this rule. Thus R is not a relation. Hence it's neither injective nor surjective and has no inverse.




(II) Since for every "a, b\\in A, ~~ R(a)=R(b)\\implies a=b\\\\ \\text{Thus, } R \\text{ is injective}\\\\ \\text{Also, every element of B is used up in the relation. Thus, } R \\text{ is surjective.}\\\\ \\text{The inverse of R is}\\\\ R^{-1}=\\{(2,1), (3,2), (4,3), (5,4), (1,5)\\}."


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