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Set Operations

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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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Note: remember all that stuff about implication?

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Union of two sets Give me the set of elements, x where x is in A or x is in B Example 0 0 0 0 1 1 1 0 1 1 1 1 OR A membership table

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Union of two sets

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

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Intersection of two sets Give me the set of elements, x where x is in A and x is in B Example 0 0 0 0 1 0 1 0 0 1 1 1 AND

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Disjoint sets

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Difference of two sets Give me the set of elements, x where x is in A and x is not in B Example 0 0 0 0 1 0 1 0 1 1 1 0

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Note: Compliment of a set

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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 0 0 0 0 1 1 1 0 1 1 1 0 XOR

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

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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|

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Cardinality of a Set The principle of inclusion-exclusion

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

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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!

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

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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

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

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

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prove using membership table Class

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Me They are the same

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prove using set builder and logical equivalence Class

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Me

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Prove using set builder and logical equivalences

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

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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

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Collections of sets

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Set Operators Goals Show how set identities are established Introduce some important identities.

Set Operators Goals Show how set identities are established Introduce some important identities.

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