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) is both symm etric and anti-symmetric. This means if then if are distinct. This is impossible. Hence only elements of are and hence Conversely , by definition, if are distinct. Hence is symmetric and anti-symmetric vacuously true.
b) Let be transitive. Let . Then is transitive. But Hence the given question is wrong(only if part).
(if part) Let Then by definition Hence Hence the relation is transitive.