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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**Union of two sets**Give me the set of elements, x where x is in A or x is in B Example A membership table 0 0 0 0 1 1 1 0 1 1 1 1 OR**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 0 0 0 0 1 0 1 0 0 1 1 1 AND**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**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**Complement of a set**Give me the set of elements, x where x is not in A Example U is the “universal set” Not**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 Set**The principle of inclusion-exclusion U Potentially counted twice (“over counted”)**Set Identities**Note similarity to logical equivalences! Identity Domination Indempotent • Think of • U as true (universal) • {} as false (empty) • Union as OR • Intersection as AND • Complement as NOT**Set Identities**Note similarity to logical equivalences! Commutative Associative Distributive De Morgan**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**Class**prove using membership table**Me**They are the same**Class**prove using set builder and logical equivalence**A containment proof**See the text book That’s a cop out if ever I saw one!**A containment proof**Guilt 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