Sets, Relation and Functions

Algebra Level 3

If R is a subset of A×B and S is the subset of B×C be two relations, then (SoR)^-1 is equal to

R^-1 oS^-1 S^-1 oR^-1 SoR RoS

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1 solution

Akil Mahmod Tipu
Jun 8, 2015

Let, ( c , a ) (c, a ) a element of ( S R ) 1 (S \circ R)^-1 where c C c \in C and a A a \in A .

So, ( c , a ) ( S R ) 1 ( a , c ) S R (c, a)\in (S \circ R)^-1 \iff (a, c) \in S \circ R b B : ( a , b ) R a n d ( b , c ) S \iff \exists b \in B: (a, b) \in R \quad and \quad (b, c) \in S b B : ( b , a ) R 1 a n d ( c , b ) S 1 \iff \exists b \in B : (b, a) \in R^-1 \quad and \quad (c, b) \in S^-1 b B : ( c , b ) S 1 a n d ( b , a ) R 1 \iff \exists b \in B : (c, b) \in S^-1 \quad and \quad (b, a) \in R^-1 ( c , a ) R 1 S 1 \iff (c, a) \in R^-1 \circ S^-1

Hence, ( S R ) 1 = R 1 S 1 (S \circ R)^-1 = R^-1 \circ S^-1 .

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