CS 104: Discrete Mathematics

1. CS 104: Discrete Mathematics Chapter 3: Fundamental Structures

2. Sets (In Book: Chapter 2-sec 2.1) T. Mai Al-Ammar

3. Sets • Set is one of the discrete structures used to represent discrete objects. • Many important discrete structures are built using sets. • Set Definition: • A set is an unordered collection of objects, called the elements or membersof the set. • A set is said to contain its elements. a denotes that a in an element of the set A. denote that a is not an element of the set A. • Although sets are usually used to group together elements with common properties, there is nothing that prevents a set from having unrelatedelements , e.g. { a, 2, Ahmed, Riyadh}. T. Mai Al-Ammar

4. Sets ( Cont. ) • Ways to describe sets : • Roster method: List all members of a set ,if possible, between braces. E.g. • S = { 1, 2, 3} • S = { 1, 3,5,9 … 99 } (( ellipses used )) • Set builder method: characterize all elements in the set by stating the property or properties they must have to be members . • e.g.S contains all odd positive integers less than 100 • N = { 0,1,2,3 , … } Natural numbers • Z = { … , -2, -1, 0, 1, 2, … } integers • Q = { p/q | rational numbers T. Mai Al-Ammar

5. Sets (Cont.) • Sets can have other sets are members , e.g. S = { N, Z, R } • In sets , the order of elements is not important since sets are unordered. • In sets, the repetition is irrelevant. • The concept of datatypein computer science is built upon the concept of a set, Datatype is the name of a set together with a set of operations that can performed on objects of the set. • E.g. Data type : booleanis the name of set {0,1} and operation on it is • { AND, OR, XOR}. T. Mai Al-Ammar

6. Sets ( Cont. ) • Equal Sets : • Two sets are equal if and only if they have the same elements, if A and B are sets, then A = B if and only if • e.g. { 1, 2,3 } = { 2 , 3, 1} • { 1 , 2, 2, 3 } = { 1,2, 3 ,3 } • Empty Set ( null set) : • A set that has no elements. It is denoted by or { } . • e.g. a set that contains positive integers less than zero • Singleton Set: • A set with one element. • Empty set is not equal to {} (( singleton set)). T. Mai Al-Ammar

7. Sets (Cont. ) • Venn Diagram : • Sets can be represented graphically using Venn diagrams. • In Venn diagram, the universal set is represented as a rectangle. • Inside rectangle, circles are used to represent sets. • And points represent elements. U V a T. Mai Al-Ammar

8. Sets (subsets ) • The set A is said to be a subsetof the set B if and only if every element of A is also an element of B, and B is said to be supersetof A. • is true • To show that A B, find a single element x A such that x B • For every set S: and • Proper subset: • if a set A is a subset of B but A B then A B. • for A B, it must be the case that is true and there must exist an element x of B that is not an element of A: • i.e. x ∉ A) • A = B if and i.e. T. Mai Al-Ammar

9. Sets ( Cont.) • Examples: • Is {x}  {x}? No • Is {x}  {x}? Yes • Is {x}  {x, {x}}? Yes • Is {x}  {x, {x}}? Yes • Size of a Set: • Let S be a set. If there are exactly n distinct elements in S where n is a nonnegative integer. Then S is a finite set and n is the cardinality of S :|S| • e.g. A is a set of odd positive integers less than 10 • A = { 1, 3, 5, 7, 9 } , |A| = 5 • || = 0 (( has no elements )) T. Mai Al-Ammar

10. Sets (power set) • Power Set : • If S is a set, then the power set of Sis the set of all subsets of the set S. • The power set of S is denoted by P(S). i.e., P(S)= { x | x  S }. • Examples: • If S={a}, then P(S)={, {a}}. • If S = {a, b}, then P(S)={, {a}, {b}, {a, b}}. • If S = , then P(S)= {} • If S = {{}}, then P(S)= {, {{}} } Fact: If S is finite, |P(S)| = 2|S| . (if |S|=n, |P(S)|=2n) T. Mai Al-Ammar

11. Sets (Cartesian Products) • The ordered n-tuples ( a1, a2, … an) is the ordered collection that has a1 as its first element and an as its nth element. • Two ordered n-tuple are equal if and only if each corresponding pair of their element is equal. • The Cartesian product of A and B, denoted by A x B , is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B : • Examples: • If A={1, 2}, B={a, b, c} then A x B={(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)} • Facts: • 1. A x B ≠ B x A 2. |A x B| = |A| * |B| • A1 x A2 x … x An = {(a1, a2,…, an) | ai  Ai, for i=1, 2, …, n} T. Mai Al-Ammar

12. Set Operations (In Book: Chapter 2-sec 2.2) T. Mai Al-Ammar

13. Sets Operations Union Intersection Disjoint Difference Complement Set Identities T. Mai Al-Ammar

14. Set operations( Cont.) 1. Union : The unionof two sets A and B, denoted by AB, is the set that contains those elements that are either in A or in B, or in both: A  B = { x | x  A or x  B} e.g. The union of two sets {1,3,5} and { 1,2,3} is { 1, 2, 3,5, } In the figure, the colored area (1,2, and 3) is the union of two sets. T. Mai Al-Ammar

15. Set operations ( Cont.) 2. Intersection : The intersectionof two sets A and B, denoted by AB, is the set containing those elements in both A and B: A  B = { x | x  A and x  B} e.g. the intersection of two sets {1,3,5 } and {1,2,3} is {1,3} In the figure, the colored area (2) is the intersection of two sets. T. Mai Al-Ammar

16. Sets Operations (Cont.) • 3. Disjoint sets: • Two sets are said to be disjointif their intersection is the empty set. • Examples: • If A = {1, 3, 5, 7, 9}, and B = {2, 4, 6, 8, 10}, then A  B=Ø • If A = {x | x is a CS major student} and B = {x | x is a IS major student}, then AB=Ø T. Mai Al-Ammar

17. Sets Operations (Cont.) 4. Difference : The differenceof two sets A and B, denoted by A-B, is the set containing those elements that are in A but not in B: A – B = { x | xA and xB} e.g. If A = {1, 3, 5}, and B = {1, 2, 3}, then A-B = {5}, and B-A = {2}. In the figure, the colored area (1) is the difference of two sets. In general : A-B  B-A T. Mai Al-Ammar

18. Set Operations (Cont.) • 5. Complement : • The Complementof the set A : is the complement of A with respect to the universal set U, i.e. those elements that are in universal set but not in A . • Therefore, the complement of the set A is U – A • e.g. If U={1, 2, …, 10}, A={1, 3, 5, 7, 9}, then ={2, 4, 6, 8, 10}. • In the figure, the colored area (3,4) is the complement of the first set. T. Mai Al-Ammar

19. Set Operations ( Cont.) 6. Set Identities T. Mai Al-Ammar

20. Set Operations (Cont.) 6. Set Identities ( Cont.) : T. Mai Al-Ammar

21. Set Operations (Cont.) Facts: = and = U A - B = A  There are three ways to proof set identities : Constructing new logical equivalence statements Use membership tables: To indicate that an element is in a set , a 1 is used, to indicate that an element is not in a set , a 0 is used. Venn diagram T. Mai Al-Ammar

22. Set Operations (Cont.) Example : Use set builder and logical equivalences to establish the first De Morgan Law T. Mai Al-Ammar

23. Set Operations (Cont.) Example: Prove the following law using Membership table: A  (B  C) = (A  B)  (A  C) T. Mai Al-Ammar

24. Set Operations (Cont.) Example: Prove the following law using Venn diagram: A  (B  C) = (A  B)  (A  C) A  (B  C) (A  B)  (A  C) T. Mai Al-Ammar

25. Set Operations (Cont.) • Remark: • Note that |A|+|B| counts each element that is in A but not in B, or in B not in A, exactly once. • Each element that is in both A and B will be counted twice • So, elements in A  B will be subtracted the result, i.e., • |A  B| = |A| + |B| - |A  B| Example: There are 150 CS majors 100 are taking CS530 , 70 are taking CS520 30 are taking both, How many are taking neither? • 150 – (100 + 70 - 30) = 10 T. Mai Al-Ammar