Set Theory: Laws and Proofs

1 Set theory : Laws and ProofsIan LuddenIan LuddenSet theory : Laws and Proofs1 / 7 Learning ObjectivesBy the end of this lesson, you will be able to: Remember fundamental laws/rules of set theory . Apply definitions and laws to set theoretic LuddenSet theory : Laws and Proofs2 / 7 Learning ObjectivesBy the end of this lesson, you will be able to: Remember fundamental laws/rules of set theory . Apply definitions and laws to set theoretic LuddenSet theory : Laws and Proofs2 / 7 Learning ObjectivesBy the end of this lesson, you will be able to: Remember fundamental laws/rules of set theory .

2 Apply definitions and laws to set theoretic LuddenSet theory : Laws and Proofs2 / 7 Set theory Properties/Identities/Laws Commutative, associative Distributive Double complement De Morgan s Laws: S T=S T S T=S T And many LuddenSet theory : Laws and Proofs3 / 7 Set theory Properties/Identities/Laws Commutative, associative Distributive Double complement De Morgan s Laws: S T=S T S T=S T And many LuddenSet theory : Laws and Proofs3 / 7 Set theory Properties/Identities/Laws Commutative, associative Distributive Double complement De Morgan s Laws: S T=S T S T=S T And many LuddenSet theory : Laws and Proofs3 / 7 Set theory Properties/Identities/Laws Commutative, associative Distributive Double complement De Morgan s Laws: S T=S T S T=S T And many LuddenSet theory : Laws and Proofs3 / 7 Set theory Properties/Identities/Laws Commutative, associative Distributive Double complement De Morgan s Laws: S T=S T S T=S T And many LuddenSet theory : Laws and Proofs3 / 7 Set theory Properties/Identities/Laws Commutative, associative Distributive Double complement De Morgan s Laws.

3 S T=S T S T=S T And many LuddenSet theory : Laws and Proofs3 / 7 Set theory Properties/Identities/Laws Commutative, associative Distributive Double complement De Morgan s Laws: S T=S T S T=S T And many LuddenSet theory : Laws and Proofs3 / 7 Cardinality after Set Operations Size of set union Size of Cartesianproduct (product rule)MenuAppetizerEntreeDessertWingsPizz aGelatoMozz. sticksPastaRhubarb PieOnion ringsSteakChoc. cakeSaladChickenCheesecakeCalamariCookie SoupIan LuddenSet theory : Laws and Proofs4 / 7 Cardinality after Set Operations Size of set union Size of Cartesianproduct (product rule)MenuAppetizerEntreeDessertWingsPizz aGelatoMozz.

4 SticksPastaRhubarb PieOnion ringsSteakChoc. cakeSaladChickenCheesecakeCalamariCookie SoupIan LuddenSet theory : Laws and Proofs4 / 7 Proving Set Inclusion A B a A,a B Leta Abe arbitrary. [Details] Soa B. Sinceawas arbitrarily chosen, we concludeA B. ExampleDefineA={a Z:a2 9is odd and|a|<25}andB={b Z:bis even}. ProveA prove set equality, show inclusion in both directionsIan LuddenSet theory : Laws and Proofs5 / 7 Proving Set Inclusion A B a A,a B Leta Abe arbitrary. [Details] Soa B. Sinceawas arbitrarily chosen, we concludeA B. ExampleDefineA={a Z:a2 9is odd and|a|<25}andB={b Z:bis even}.

5 ProveA prove set equality, show inclusion in both directionsIan LuddenSet theory : Laws and Proofs5 / 7 Proving Set Inclusion A B a A,a B Leta Abe arbitrary. [Details] Soa B. Sinceawas arbitrarily chosen, we concludeA B. ExampleDefineA={a Z:a2 9is odd and|a|<25}andB={b Z:bis even}. ProveA prove set equality, show inclusion in both directionsIan LuddenSet theory : Laws and Proofs5 / 7 Proving Set Inclusion A B a A,a B Leta Abe arbitrary. [Details] Soa B. Sinceawas arbitrarily chosen, we concludeA B. ExampleDefineA={a Z:a2 9is odd and|a|<25}andB={b Z:bis even}. ProveA prove set equality, show inclusion in both directionsIan LuddenSet theory : Laws and Proofs5 / 7 Proving Set Inclusion A B a A,a B Leta Abe arbitrary.

6 [Details] Soa B. Sinceawas arbitrarily chosen, we concludeA B. ExampleDefineA={a Z:a2 9is odd and|a|<25}andB={b Z:bis even}. ProveA prove set equality, show inclusion in both directionsIan LuddenSet theory : Laws and Proofs5 / 7 Proving Set Inclusion A B a A,a B Leta Abe arbitrary. [Details] Soa B. Sinceawas arbitrarily chosen, we concludeA B. ExampleDefineA={a Z:a2 9is odd and|a|<25}andB={b Z:bis even}. ProveA prove set equality, show inclusion in both directionsIan LuddenSet theory : Laws and Proofs5 / 7 Proving Set Inclusion A B a A,a B Leta Abe arbitrary. [Details] Soa B.

7 Sinceawas arbitrarily chosen, we concludeA B. ExampleDefineA={a Z:a2 9is odd and|a|<25}andB={b Z:bis even}. ProveA prove set equality, show inclusion in both directionsIan LuddenSet theory : Laws and Proofs5 / 7 Another Set ProofLetA,B,C U. Prove that(A B) Cif and only if(A C) LuddenSet theory : Laws and Proofs6 / 7 Learning ObjectivesBy the end of this lesson, you will be able to: Remember fundamental laws/rules of set theory . Apply definitions and laws to set theoretic of set theory laws: LuddenSet theory : Laws and Proofs7 / 7