Relations-Properties and Examples (p3op2)op1=p3o(p2o P1) polds Cp2ldTop dom(Ids)=S=ran(Ids) IdTo lds =ldTns Ids-1=Ids (p-1)-1=p (p2op1)-1=p1-1op2-1 po0=0=0op 1dg=0=0-1 dom(p)=0←→p=0 14/40 Relations – Properties and Examples (ρ3 ◦ ρ2) ◦ ρ1 = ρ3 ◦ (ρ2 ◦ ρ1) ρ ◦ IdS ⊆ ρ ⊇ IdT ◦ ρ dom(IdS ) = S = ran(IdS ) IdT ◦ IdS = IdT∩S IdS −1 = IdS ￾ ρ −1
−1 = ρ (ρ2 ◦ ρ1) −1 = ρ1 −1 ◦ ρ2 −1 ρ ◦ ∅ = ∅ = ∅ ◦ ρ Id∅ = ∅ = ∅ −1 dom(ρ) = ∅ ⇐⇒ ρ = ∅ 14 / 40