350 likes | 591 Views
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.
E N D
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