Week 12 Properties of Relations

Week 12 Properties of Relations - R β—¦ R βŠ† R Proof R is...

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CONCORDIA UNIVERSITY COMP 232/2 Mathematics for Computer Science FALL 2010 Properties of Relations Reflexive x : x Symmetric x,y : x y -→ x y Antisymmetric x,y : x y -→ x=y Transitive x,y,z : x y z -→ x y z Let R be a relation on a set A . Proposition 1. R is reflexive if and only if Δ A R . Proof. R is reflexive ≡ ∀ a ± ( a, a ) R ² ≡ ∀ a b ± a = b ( a, b ) R ² ≡ ∀ a b ± ( a, b ) Δ A ( a, b ) R ² Δ A R. Proposition 2. R is symmetric if and only if R - 1 R . Proof. R is symmetric ≡ ∀ a b ± ( a, b ) R ( b, a ) R ² ≡ ∀ a b ± ( b, a ) R - 1 ( b, a ) R ² R - 1 R. Proposition 3. R is antisymmetric if and only if R R - 1 Δ A . Proof. R is antisymmetric ≡ ∀ a b ± ( a, b ) R ( b, a ) R a = b ² ≡ ∀ a b ± ( a, b ) R ( b, a ) R ( a, b ) Δ A ² ≡ ∀ a b ± ( a, b ) R ( a, b ) R - 1 ( a, b ) Δ A ² ≡ ∀ a b ± ( a, b ) R R - 1 ( a, b ) Δ A ² R R - 1 Δ A . Proposition 4. R is transitive if and only if
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Unformatted text preview: R β—¦ R βŠ† R . Proof. R is transitive ≑ βˆ€ a βˆ€ b βˆ€ c Β± ( a, b ) ∈ R ∧ ( b, c ) ∈ R β†’ ( a, c ) ∈ R Β² ≑ βˆ€ a βˆ€ b βˆ€ c Β± ( a, b ) / ∈ R ∨ ( b, c ) / ∈ R ∨ ( a, c ) ∈ R Β² ≑ βˆ€ a βˆ€ c Β± βˆ€ b [( a, b ) / ∈ R ∨ ( b, c ) / ∈ R ∨ ( a, c ) ∈ R ] Β² ≑ βˆ€ a βˆ€ c Β± βˆ€ b [( a, b ) / ∈ R ∨ ( b, c ) / ∈ R ] ∨ ( a, c ) ∈ R Β² ≑ βˆ€ a βˆ€ c Β± Β¬βˆƒ b [( a, b ) ∈ R ∧ ( b, c ) ∈ R ] ∨ ( a, c ) ∈ R Β² ≑ βˆ€ a βˆ€ c Β± βˆƒ b [( a, b ) ∈ R ∧ ( b, c ) ∈ R ] β†’ ( a, c ) ∈ R Β² ≑ βˆ€ a βˆ€ c Β± ( a, c ) ∈ R β—¦ R β†’ ( a, c ) ∈ R Β² ≑ R β—¦ R βŠ† R. 9-Sep-2010, 4:02 pm...
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This note was uploaded on 11/15/2010 for the course ENCS COMP 232 taught by Professor Ford during the Fall '10 term at Concordia University Irvine.

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