2. Chapter 5 Section 1 Sets

3. Basics Set –Definition: Collection of objects –Specified by listing the elements of the set inside a pair of braces. –Denoted by a capital letter Elements –Objects in a set –Common set elements: Numbers, Names, Letters, Years, etc.

4. Universal Set and Subsets Universal set (U): –Contains all the sets that we are interested in –All the elements that we are considering are contained in the universal set Subset –Sets formed by selecting elements from a set (usually the universal set)

5. Exercise 1 (page 210) U = { 1, 2, 3, 4, 5, 6, 7 } S = {1, 2, 3, 4 } T = { 1, 3, 5, 7 } List elements in the following sets (a) S´,(b) S U T, (c) S ∩ T, (d) S´ ∩ T

6. Complement Set Definition: The set of elements of U that are not in set A is called the compliment set of A, denoted by A´ Other notations occasionally used: A c or A Exercise 1 (a) List S´ U = { 1, 2, 3, 4, 5, 6, 7 }, S = {1, 2, 3, 4 } Answer: S´ = { 5, 6, 7 }

7. Union of two sets The union of two sets ( A and B ) consists of all elements that are in A or B or both Notation: A U B Exercise 1 (b) List S U T S = {1, 2, 3, 4 }, T = { 1, 3, 5, 7 } Answer: S U T = { 1, 2, 3, 4, 5, 7 }

8. Intersection of Two Sets The intersection of two sets ( A and B ) consists of all elements that are in both A and B Notation: A ∩ B Exercise 1 (c) List S ∩ T S = {1, 2, 3, 4 }, T = { 1, 3, 5, 7 } Answer: S ∩ T = { 1, 3 }

9. Empty / Null Set Set that contains no elements Denoted by either:{ } or Ø Note that { Ø } is NOT a notation for the empty set. This is a common mistake that is made.

10. Important Relationships 1.S ∩ S´ = Ø 2.S U S´ = U 3.A set is always a subset of itself 4.Ø is always a subset of any set

11. Exercise 13 (page 210) Part (a) U = { a, b, c, d, e, f } R = { a, b, c }, S = { a, c, e }, T = { e, f } Part (a): List ( R U S ) ´ R U S = { a, b, c, e } ( R U S ) ´ = { d, f }

12. Exercise 13 Part (d) R = { a, b, c }, S = { a, c, e }, T = { e, f } Part (d): List R ∩ S ∩ T ´ T ´= { a, b, c, d } R ∩ S = { a, c } R ∩ S ∩ T´ = { a, c } ∩ T ´ = { a, c } ∩ { a, b, c, d } = { a, c } R ∩ S ∩ T ´ = { a, c }

13. Exercise 13 Part (g) R = { a, b, c }, S = { a, c, e }, T = { e, f } Part (g): List ( R U S ) ∩ ( R U T ) ( R U S ) = { a, b, c, e } ( R U T ) = { a, b, c, e, f } ( R U S ) ∩ ( R U T ) = { a, b, c, e } ∩ { a, b, c, e, f } ( R U S ) ∩ ( R U T ) = { a, b, c, e }