3a-Set theory.ppt
- 1. 9/21/2023 CS 201 1 CS201: Data Structures and Discrete Mathematics I Basic Set Theory
- 2. 9/21/2023 CS 201 2 Sets • A set is a collection of distinct objects. • For example (let A denote a set): A = {apple, orange, grape} A = {1, 2, 3, 4, 5} A = {1, b, c, d, e, f} A = {(1, 2), (3, 4), (9, 10)} A = {<1, 2, 3>, <3, 4, 5>, <6, 7, 8>} A = a collection of anything that is meaningful.
- 3. 9/21/2023 CS 201 3 Members and Equality of Sets • The objects that make up a set are called members or elements of the set. • Two sets are equal iff they have the same members. – That is, a set is completely determined by its members. – Order and repetition do not matter in a set.
- 4. 9/21/2023 CS 201 4 Set notations • The notation of {...} describes a set. Each member or element is separated by a comma. – E.g., S = {apple, pear, grape} – S is a set – The members of S are: apple, pear, grape • Order and repetition do not matter in a set. • The following expressions are equivalent: – {1, 3, 9} – {9, 1, 3} – {1, 1, 3, 3, 9}
- 5. 9/21/2023 CS 201 5 The membership symbol and the empty set • The fact that x is a member of a set S can be expressed as – x S – Reads: • x is in S, or • x is a member of S, or • X belongs to S • An example, S = {1, 2, 3}, 1 S, 2 S, 3 S • The negation of is written as (is not in). • The empty set is a set without any element – Denoted by {} or – For any object x, x
- 6. 9/21/2023 CS 201 6 Defining a Set by membership properties • Notation o S = {x T | P(x)} (or S = {x | x T and P(x)}) o The members of S are members of an already known set T that satisfy property P. • An example: o Let Z be the set of integers o Let Z+ be the set of positive integers. o Z+ = {x Z | x > 0}
- 7. 9/21/2023 CS 201 7 Sets of numbers • Z = The set of all integers Z = {…, -2, -1, 0, 1, 2, …} • Z+ = The set of positive integers Z+ = {1, 2, 3…} = {x | x Z and x > 0} = {x Z | x > 0} • Z- = The set of negative integers Z- = {…, -3, -2, -1} = {-1, -2, -3…} = {x Z | x < 0} • R = The set of all real numbers • Q = the set of all rational numbers Q = {x R | x = p/q and p, q Z and q 0} • We can use “;” to replace “and”
- 8. 9/21/2023 CS 201 8 Subsets • A is a subset () of B, or B is a superset of A iff every member of A is a member of B. o A B iff for all x if x A, then x B • An example: o (-2, 0, 6} {-3, -2, -1, 0, 1, 3, 6} • Negation: A is not a subset of B or B is not a superset of A iff there is a member of A that is not a member of B o A B iff there exist x, x A, x B
- 9. 9/21/2023 CS 201 9 Obvious subsets – S S – S
- 10. 9/21/2023 CS 201 10 Proper subsets • A is a proper subset () of B, or B is a proper superset of A iff A is a subset of B and A is not equal to B. o A B iff A B and A B • Examples: o {1, 2, 3} {1, 2, 3, 4, 5} o Z+ Z Q R o If S then S
- 11. 9/21/2023 CS 201 11 Power sets • The set of all subsets of a set is called the power set of the set • The power set of S is denoted by P(S). • Example: – P() ={} – P({1, 2}) = {, {1}, {2}, {1, 2}} – P(S) = {, …, S} – What is P({1, 2, 3})? • How many elements does the power set of S have? Assume S has n elements. 2 ^ n
- 12. 9/21/2023 CS 201 12 and are different • Examples: 1 {1} is true 1 {1} is false {1} {1} is true • Which of the following statement is true? S P(S) S P(S) The 1st is true
- 13. 9/21/2023 CS 201 13 Mutual inclusion and set equality • Sets A and B have the same members iff they mutually include – A B and B A • That is, A = B iff A B and B A • Mutual inclusion is very useful for proving the equality of sets • To prove an equality, we prove two subset relationships.
- 14. 9/21/2023 CS 201 14 Universal sets • Depending on the context of discussion – Define a set of U such that all sets of interest are subsets of U. – The set U is known as a universal set • Examples: – When dealing with integers, U may be Z. – When dealing with plane geometry, U may be the set of points in the plane
- 15. 9/21/2023 CS 201 15 Venn diagram • Venn diagram is used to visualize relationships of some sets. • Each subset (of U, the rectangle) is represented by a circle inside the rectangle.
- 16. 9/21/2023 CS 201 16 Set operations • Let A, B be subsets of some universal set U. • The following set operations create new sets from A and B. • Union: A B = {x U | x A or x B} • Intersection: A B = {x U | x A and x B} • Difference: A B = A B= {x U | x A and x B} • Complement A = U A = {x U | x A}
- 17. 9/21/2023 CS 201 17 Set union • An example {1, 2, 3} {1, 2, 4, 5} = {1, 2, 3, 4, 5} The venn diagram 1 2 3 4 5
- 18. 9/21/2023 CS 201 18 Set intersection • An example {1, 2, 3} {1, 2, 4, 5} = {1, 2} The venn diagram 1 2 3 4 5
- 19. 9/21/2023 CS 201 19 Set difference • An example {1, 2, 3} - {1, 2, 4, 5} = {3} The venn diagram 1 2 3 4 5
- 20. 9/21/2023 CS 201 20 Set complement • The venn diagram
- 21. 9/21/2023 CS 201 21 Basic set identities (equalities) • Commutative laws A B = B A A B = B A • Associative laws (A B) C = A (B C) (A B) C = A (B C) • Distributive laws A (B C) = (A B) (A C) A (B C) = (A B) (A C)
- 22. 9/21/2023 CS 201 22 Basic set identities (cont …) • Identity laws A = A = A A U = U A = A • Double complement law (A’)’ = A • Idempotent laws A A = A A A = A • De Morgan’s laws (A B)’ = A’ B’ (A B)’ = A’ B’
- 23. Basic set identities (cont …) • Absorption laws A (A B) = A A (A B) = A • Complement law (U)’ = ’ = U • Set difference law A – B = A B’ • Universal bound law A U = U A = 9/21/2023 CS 201 23
- 24. 9/21/2023 CS 201 24 Infinite sets • In a finite set, we can always designate one element as the first member, s1, another element as the second member, s2 and so on. If there are k elements in the set we can list them as – s1, s2, …, sk • A set that is not finite is called an infinite set. • If a set is infinite, we may still be able to select a first element s1, a second element s2 and so on: – s1, s2, … • Both above sets are countable. • Countable does not mean we can give a total number, but means that we can say, “here is the first one” and “here is the second one” and so on.
- 25. 9/21/2023 CS 201 25 Countable sets: examples • The set of positive integer numbers are countable. • The set of positive rational numbers are countable
- 26. 9/21/2023 CS 201 26 Uncountable sets • There are also some sets that are uncountable. – The set is so large, and there is no way to count out the elements. • One example: The set of real numbers between 0 and 1 is uncountable. • A computer can only manage finite sets.
- 27. 9/21/2023 CS 201 27 Summary • Sets are extremely important for Computer Science. • A set is simply an unordered list of objects. • Set operations: union, intersection, difference.