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math 61

# math 61 - R op by the following R op = a b b a ∈ R Show...

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Hey guys here a solutions to some problems that a student asked me to email out. Problem 1 Show that if R and S are transitive then R S is transitive. Proof Let (a,b) R S and (b,c) R S . Then to show that R S is transitive we need to show that (a,c) R S . Now (a,b) R S means that (a,b) R and (a,b) S . Similarly we have that (b,c) R and (b,c) S . But R and S are both transitive. Thus (a,c) R and (a,c) S . Thus (a,c) R S . Problem 2 Given a relation R define a new relation
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Unformatted text preview: R op by the following: R op = { ( a, b ) : ( b, a ) ∈ R } Show that if R is transitive then R op is also transitive. Proof Let (a,b) ∈ R op and (b,c) ∈ R op . As above we need to show that (a,c) ∈ R op . Now (a,b) ∈ R op means that (b,a) ∈ R . Similarly (c,b) ∈ R . Thus since R is transitive this means that (c,a) ∈ R . But this means that (a,c) ∈ R op . So we are done. 1...
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