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# lec1019 - Problems involving relation composition Prove or...

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Problems involving relation composition Prove or disprove: If a relation R, defined as a subset of A x A, is symmetric, then show that R ° R is symmetric as well. Since R is symmetric, we know that if (a,b) R, then (b,a) R. We must prove under this assumption that if (a,b) R ° R, then (b,a) R ° R. If (a,b) R ° R, by the definition of function composition, we must have an element c A such that (a,c) R and (c,b) R. But we know R is symmetric. Thus, we can deduce that (c,a) R and (b,c) R. But, if this is the case, by the definition of function composition, we must have (b,a) R ° R, because (b,c) R and (c,a) R. This is exactly what we needed to prove. Thus, R ° R is symmetric. Prove: R is transitive R ° R R. We must show that if (a,b) R ° R, then (a,b) R. First of all, if (a,b) R ° R, then by the definition of function composition, there exists an element c of A such that (a,c) R and (c,b) R.

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