Set Operations. When sets are equal A equals B iff for all x, x is in A iff x is in B or … and this is what we do to prove sets equal.

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Presentation transcript:

Set Operations

When sets are equal A equals B iff for all x, x is in A iff x is in B or … and this is what we do to prove sets equal

Note: remember all that stuff about implication?

Union of two sets Give me the set of elements, x where x is in A or x is in B Example OR A membership table

Union of two sets

Note: we are using set builder notation and the laws of logical equivalence (propositional equivalence)

Intersection of two sets Give me the set of elements, x where x is in A and x is in B Example AND

Disjoint sets

Difference of two sets Give me the set of elements, x where x is in A and x is not in B Example

Note: Compliment of a set

Symmetric Difference of two sets Give me the set of elements, x where x is in A and x is not in B OR x is in B and x is not in A Example XOR

Complement of a set Give me the set of elements, x where x is not in A Example Not U is the universal set

Cardinality of a Set In claire A = {1,3,5,7} B = {2,4,6} C = {5,6,7,8} |A u B| ? |A u C| |B u C| |A u B u C|

Cardinality of a Set The principle of inclusion-exclusion

Cardinality of a SetThe principle of inclusion-exclusion U Potentially counted twice (over counted)

Set Identities Identity Domination Think of U as true (universal) {} as false (empty) Union as OR Intersection as AND Complement as NOT Indempotent Note similarity to logical equivalences!

Set Identities Commutative Associative Distributive De Morgan Note similarity to logical equivalences!

Four ways to prove two sets A and B equal a membership table a containment proof show that A is a subset of B show that B is a subset of A set builder notation and logical equivalences Venn diagrams

Prove lhs is a subset of rhs Prove rhs is a subset of lhs

… set builder notation and logical equivalences Defn of complement Defn of intersection De Morgan law Defn of complement Defn of union

prove using membership table Class

Me They are the same

prove using set builder and logical equivalence Class

Me

Prove using set builder and logical equivalences

A containment proof See the text book Thats a cop out if ever I saw one!

A containment proofGuilt kicks in To do a containment proof of A = B do as follows 1. Argue that an arbitrary element in A is in B i.e. that A is an improper subset of B 2. Argue that an arbitrary element in B is in A i.e. that B is an improper subset of A 3. Conclude by saying that since A is a subset of B, and vice versa then the two sets must be equal

Collections of sets