PDF - Computer Science - Brock University
PDF - Computer Science - Brock University
PDF - Computer Science - Brock University
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2. Property (1) shows ’‘⊑”. We obtain the other inclusion from<br />
M(β,γ) ⊑ M(β,γ); M(β,γ) ⌣ ; M(β,γ)<br />
= M(β,γ); M ⌣ (γ,β); M(β,γ)<br />
= M(β,γ); M(γ,β); M(β,γ) M symmetric<br />
⊑ M(β,β); M(β,γ) by (1)<br />
3. From (1) we obtain M(γ,β); M(β,δ); R ⊑ M(γ,δ); R. This implies the assertion<br />
because M is symmetric.<br />
⊓⊔<br />
The next lemma allows a stepwise definition of partial equivalence relations<br />
based on the diagonal elements of M.<br />
Lemma 5. If M : f → f is a partial equivalence relation in R + of size α, then<br />
the relations<br />
Ξ β := M(β,β) ⊓<br />
γ
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