Solution to Let ρ be a relation on a set A. Define ρ −1 = { | … - Sikademy
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Archangel Macsika

Let ρ be a relation on a set A. Define ρ −1 = { | ∈ ρ}. Also for two relations ρ, σ on A, define the composite relation ρ ◦ σ as (a, c) ∈ ρ ◦ σ if and only if there exists b ∈ A such that ∈ ρ and ∈ σ. Prove the following assertions (a) ρ is both symmetric and antisymmetric if and only if ρ ⊆ { | a ∈ A}. (b) ρ is transitive if and only if ρ ◦ ρ = ρ.

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a) \rho is both symm etric and anti-symmetric. This means if (a,b)\in \rho then (b,a)\in \rho , (b,a)\notin\rho if a,b are distinct. This is impossible. Hence only elements of \rho are (a,a) and hence \rho\subseteq\{(a,a)|a\in A\}. Conversely , by definition, (a,b)\notin\rho if a,b are distinct. Hence \rho is symmetric and anti-symmetric vacuously true.

b) Let \rho be transitive. Let \rho =\{(1,2),(2,3),(1,3),(5,6)\}. Then \rho is transitive. But \rho o\rho =\{(1,3)\}\neq \rho. Hence the given question is wrong(only if part).

(if part) Let (a,b) , (b,c)\in \rho. Then by definition (a,c)\in\rho o \rho. Hence (a,c)\in \rho. Hence the relation is transitive.


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