Answer to Question #118002 in Discrete Mathematics for yousef

Question

Let ρ ⊆ S × T, for finite sets S and T. Define fρ : S −→ P(T) be such that fρ(a) = {b | aρb}.
a) Prove that ρ is reflexive if and only if a ∈ fρ(a), for every a ∈ S.
b) Prove that ρ is symmetric if and only if a ∈ fρ(b), for every a ∈ S and b ∈ fρ(a).
c) Prove that ρ is transitive if and only if fρ(b) ⊆ fρ(a), for every a ∈ S and b ∈ fρ(a).
d) Prove that ρ is not many-to-one if and only if fρ(a) ∩ fρ(b) 6= ∅ implies a = b, for every
a, b ∈ S.
e) Can fρ be onto? Explain your reasoning.

Answer 1

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