Sets Reading: EC 3.1-3.3 Peter J. Haas INFO 150 Fall Semester - PowerPoint PPT Presentation

Sets Reading: EC 3.1-3.3 Peter J. Haas INFO 150 Fall Semester 2019 Lecture 11 1/ 21

Sets Definitions Defining Sets Set Operations The Inclusion-Exclusion Principle Cartesian Products The Power Set Lecture 11 2/ 21

Some Common Sets Loose definition: A set is a collection of objects (called members or elements) I This loose general definition can lead to paradoxes (e.g., Russell’s Paradox) I To stay out of trouble, we will work with a small number of well-understood sets Basic Sets I N = { 0 , 1 , 2 , 3 , . . . } : The set of natural numbers I Z = { . . . , � 3 , � 2 , � 1 , 0 , 1 , 2 , 3 , . . . } : The set of integers I Q : The set of rational numbers, e.g., ratios of integers such as 2 3 , 3 1 , � 17 4 I R : The set of real numbers, i.e., decimal numbers with possibly infinite strings of digits after the decimal point Variations on Basic Sets I R + : The set of positive real numbers I R � 0 : The set of nonnegative real numbers I Q + : The set of positive rationals I Q � 0 : The set of nonnegative rationals I Z + : The set of positive integers I Z � 0 : The same as N Lecture 11 3/ 21

Subsets Definitions for Subsets I x 2 A : The element x is a member of the set A I A ✓ B : A is a subset of B , i.e., every element in A is also in B I A = B : Set A and B contain exactly the same elements 13 I ; : The empty set, i.e., the set that contains no elements I U : For any given discussion, all sets will be subsets of a larger set called the universal set or the universe Some Formal Definitions I A ✓ B : 8 x 2 U , ( x 2 A ) ! ( x 2 B ) I A = B : A ✓ B and B ✓ A True or False? (If false, give a counterexample) - 3) L I Z ✓ N false I N ✓ Z True I 3G ) } False I Q ✓ Z 3 e I Z ✓ Q True Ira ) False I R ✓ Q I Q ✓ R True Lecture 11 4/ 21

√ Digression: 2 is Irrational Theorem p 2 is irrational Proof by Contradiction: ( reduced form ) 1. Suppose the theorem is false p 2 = a 2. Then we can write b where a and b are relatively prime 3. So 2 = a 2 b 2 , or a 2 = 2 b 2 G 4. Therefore a 2 is even, which implies that a is even (see end of Lecture 7) 5. Therefore a = 2 k for some integer k 6. So a 2 = 4 k 2 = 2 b 2 and hence b 2 = 2 k 2 7. Therefore b 2 is even, so that b is even 8. Thus a and b are both even 9. This contradicts the assumption that a and b are relatively prime 10. Since assuming that the theorem is false leads to a contradiction, the theorem must be true. Lecture 11 5/ 21

" " duplicates contain bag More Examples a set considers ← 1. { 1 , 2 , 3 , 4 , 5 } = { 4 , 2 , 3 , 1 , 5 } = { 1 , 1 , 2 , 2 , 3 , 3 , 4 , 4 , 5 , 5 , 5 , 5 } elements only unique 2. { 2 , 4 } ✓ { 1 , 2 , 3 , 4 , 5 } is true 3. { 1 , 2 , 3 , 4 , 5 } ✓ { 2 , 4 } is false I E.g., 1 is a counterexample to “if x 2 { 1 , 2 , 3 , 4 , 5 } , then x 2 { 2 , 4 } ” 4. ; ✓ { 1 , 2 , 3 , 4 , 5 } is true I There is no counterexample to “if x 2 ; , then x 2 { 1 , 2 , 3 } ” I The empty set is a subset of every set 5. { John , Sue , Chen , Shankar } is a set containing 4 names I U = the set of all first names of people 6. { (1 , 3) , (2 , 5) , (3 , 7) } is a set of ordered pairs 7. {{ 3 , 4 } , { 5 , 6 , 7 }} is a set of sets Lecture 11 6/ 21

Set-Builder Notation Even natural numbers I { x : x 2 N and x is even } or I { x 2 N : x is even } or I { x 2 N : x = 2 k for some k 2 N } (a property description) I { 2 k : k 2 N } (a form description) closed interval Intervals intervals ° interval I { x 2 R : � 2 . 1  x  2 . 6 } or [ � 2 . 1 , 2 . 6] . open I { x 2 R : � 2 . 1 < x < 2 . 6 } or ( � 2 . 1 , 2 . 6) closed } semi I { x 2 R : � 2 . 1 < x  2 . 6 } or ( � 2 . 1 , 2 . 6] I { x 2 R : � 2 . 1  x < 2 . 6 } or [ � 2 . 1 , 2 . 6) I { x 2 N : 3  x < 6 } or [3 , 6) = { 3 , 4 , 5 } Form description prime } Other examples: give an alternate description bane . b a I { n 2 N : n has exactly two positive divisors } Sa : , ¢ I { x 2 R : x 2 + 1 = 0 } Lecture 11 7/ 21

Examples of Form Notation 0,3 6 1. The set of integers that are multiples of 3: { 3 k : k 2 Z } - u - , , 2. The set of perfect square integers: { m 2 : m 2 N } or { m 2 : m 2 Z } , 4,9 , O I - v . , 3. The set of natural numbers that end in a 1: { 10 k + 1 : k 2 N } 41121 - , - 4. The set Q : { a b : a 2 Z and b 2 Z + } Lecture 11 8/ 21

Operations on Sets Operations on Sets A and B 1. Intersection A \ B : A \ B = { x 2 U : x 2 A and x 2 B } 2. Union A [ B : A [ B = { x 2 U : x 2 A or x 2 B } 3. Di ff erence A � B : A � B = { x 2 U : x 2 A and x 62 B } 4. Complement A 0 : A 0 = { x 2 U : x 62 A } or A 0 = U � A Definition Set A and B are disjoint if A \ B = ; . Example: Suppose U = { 1 , 2 , . . . , 12 } , A = { 1 , 2 , 3 , 4 , 5 } , B = { 2 , 4 , 6 , 8 , 10 } I A 0 = , 14 { 6,7 , . . . I A \ B = 5443 , 8,10 ) , 3,6 { I I A [ B = , 43,4 , 3,53 I I I A � B = , 8,10 } I B � A = I 6 to I A \ { 8 , 10 , 12 } = 0 I U 0 = Lecture 11 9/ 21

to BHT for Sets logic Relation → false of close v U → true n u n → → Theorem For sets A , B , and C , the empty set ; , and the universal set U , the following properties hold: (a) Commutative A \ B = B \ A A [ B = B [ A (b) Associative ( A \ B ) \ C = A \ ( B \ C ) ( A [ B ) [ C = A [ ( B [ C ) (c) Distributive A \ ( B [ C ) = ( A \ B ) [ ( A \ C ) A [ ( B \ C ) = ( A [ B ) \ ( A [ C ) (d) Identity A \ U = A A [ ; = A A [ A 0 = U A \ A 0 = ; (e) Negation ( A 0 ) 0 = A (f) Double negative (g) Idempotent A \ A = A A [ A = A ( A \ B ) 0 = A 0 [ B 0 ( A [ B ) 0 = A 0 \ B 0 (h) DeMorgan’s laws (i) Universal bound A [ U = U A \ ; = ; (j) Absorption A \ ( A [ B ) = A A [ ( A \ B ) = A U 0 = ; ; 0 = U (k) Complements A � B = A \ B 0 (l) Compl. & neg. ¢ Duality principle u : s → n u u u → of → n Lecture 11 10/ 21

Verification via Venn Diagrams U A B 3 Example: U = { 1 , 2 , . . . , 15 , 16 } , A = { 1 , 2 , 5 , 7 , 9 , 11 , 13 , 15 } , 5 7 11 15 2 B = { 2 , 3 , 5 , 7 , 11 , 13 } , C = { 1 , 4 , 9 , 16 } 13 9 1 6 16 4 10 12 C 8 Example: Show that A \ ( B [ C ) = ( A \ B ) [ ( A \ C ) 14 A B A B Also: A \ ( B [ C ) ✓ B [ C C C U U A B A B A B C C C U U U Lecture 11 11/ 21

→ HEB ) Elementwise Proofs 1 e A) tlxeu :@ Definition Example 1: Prove that A ∩ B ⊆ A 1. Let x ∈ A ∩ B 2. Then x ∈ A and x ∈ B 3. In particular, x ∈ A 4. So ( x ∈ A ∩ B ) → ( x ∈ A ), and hence A ∩ B ⊆ A Lecture 11 12/ 21

An LB uld ) thou Elementwise Proofs 2 , # formal Version : ) get B) YAK ) → Example 2: Prove that A ∩ ( B ∪ C ) ⊆ ( A ∩ B ) ∪ ( A ∩ C ) 1. Let x ∈ A ∩ ( B ∪ C ) 2. Then x ∈ A and x ∈ B ∪ C 3. Case 1: x ∈ B 3.1 Then x ∈ A ∩ B since x ∈ A and x ∈ B 3.2 Hence in ( A ∩ B ) ∪ ( A ∩ C ) by argument similar to Example 1 4. Case 2: x ∈ C 4.1 Then x ∈ A ∩ C 4.2 Hence in ( A ∩ B ) ∪ ( A ∩ C ) by argument similar to Example 1 5. In either case, x ∈ A ∩ ( B ∪ C ) implies that x ∈ ( A ∩ B ) ∪ ( A ∩ C ) I Lecture 11 13/ 21

Elementwise Proofs 3 Example 3: Prove that A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) I By above result, it su ffi ces to show that ( A ∩ B ) ∪ ( A ∩ C ) ⊆ A ∩ ( B ∪ C ) not @ Let CAN B) U x a that NEA hence HEA i. show and exo B xe A then An B I :X a hence HEA case . ' and xeA C Anc Then Case 2 : exe . ' both XEA cases in so . Show that XO BVC a . XEB BUC An hence Xe > B case I xe : , GC hence X Arc Xo Laser : L BUD , 46 An Bvc hence X EA and XE 3 , - Lecture 11 14/ 21

Elementwise Proofs 4 Example 4: Prove that ( A ⊆ B ) ∧ ( B ⊆ C ) → ( A ⊆ C ) 1. Let x ∈ A 2. Then x ∈ B since A ⊆ B 3. Hence x ∈ C since B ⊆ C 4. So ( x ∈ A ) → ( x ∈ C ), and hence A ⊆ C Lecture 11 15/ 21

The Inclusion-Exclusion Principle A B Definition C n ( A ) = the number of elements in the set A U Example I A = { 2 k : k 2 Z + and k  15 } and B = { 3 k : k 2 Z + and k  10 } F 3 I n ( A ) = n ( B ) = } 2,416,410,13143^936,9 ) I A \ B = U - 163 A Any B I I n ( A \ B ) = - since 2 4 3 6 8 14 18 9 I 2,346,8 , 9,1914143 10 12 I A [ B = 16 15 24 30 20 22 21 71-3 26 28 27 I n ( A [ B ) = I - Theorem (The Inclusion-Exclusion Principle) Let sets A , B , and C be given. Then I n ( A [ B ) = n ( A ) + n ( B ) � n ( A \ B ) I n ( A [ B [ C ) = n ( A )+ n ( B )+ n ( C ) � n ( A \ B ) � n ( A \ C ) � n ( B \ C )+ n ( A \ B \ C ) n ( A ) + n ( B ) + n ( C ) = n 1 + n 2 + n 3 + 2 n 4 + 2 n 5 + 2 n 6 + 3 n 7 Lecture 11 16/ 21

Cartesian Products Ordered pairs: Definition The Cartesian product A ⇥ B of sets A and B is defined as { ( a , b ) : a 2 A and b 2 B } . When both coordinates are taken from the same set A , we write A 2 instead of A ⇥ A . Example: For second graph, the set of plotted points are { ( x , y ) 2 R 2 : y = 3 x � 1 } Example: Succinctly describe the sequence a 1 = 2 , a 2 = 4 , a 3 = 8 , . . . using ordered scn.sn ) I pairs (1 , 2) , (2 , 4) , (3 , 8), etc.: nett : Example: Describe all possible (sandwich, drink) orders where sandwiches are of type Gbh ,3)xSA,B,4D3 1, 2, or 3 and drinks are A , B , C , or D : 49,13A } . - Example: Describe all possible pairs (type of A, type of B) of people you might teller }t=E4TlxE4T3 :{ encounter on the island of Liars and Truthtellers: ( 417USD Truth Liar , , Lecture 11 17/ 21