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Introduction to sets

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S
Sonia Pahuja

This document defines basic set theory concepts including sets, elements, notation for sets, subsets, unions, intersections, and empty sets. It provides examples and notation for these concepts. Sets are collections of objects, represented with capital letters, while elements are individual objects represented with lowercase letters. Notation includes braces { } to list elements and vertical bars for set builder notation. A is a subset of B if all elements of A are also in B, written as A ⊆ B. The union of sets A and B contains all elements that are in A or B, written as A ∪ B. The intersection of sets A and B contains elements that are only in both A and B, written as A ∩ B.

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August 2006 1
August 2006 2 Definitions • A set is a collection of objects. • Objects in the collection are called elements of the set.
August 2006 3 Examples - set The collection of persons living in INDIA is a set. – Each person living in INDIA is an element of the set. The collection of all countries in the state of Texas is a set. – Each county in Texas is an element of the set.
August 2006 4 Examples - set The collection of counting numbers is a set. – Each counting number is an element of the set. The collection of pencils in your briefcase is a set. – Each pencil in your briefcase is an element of the set.
August 2006 5 Notation • Sets are usually designated with capital letters. • Elements of a set are usually designated with lower case letters.
August 2006 6 Notation • The roster method of specifying a set consists of surrounding the collection of elements with braces. For example the set of counting numbers from 1 to 5 would be written as {1, 2, 3, 4, 5}.
August 2006 7 Notation • Set builder notation has the general form {variable | descriptive statement }. The vertical bar (in set builder notation) is always read as “such that”. Set builder notation is frequently used when the roster method is either inappropriate or inadequate.
August 2006 8 Example – set builder notation {x | x < 6 and x is a counting number} is the set of all counting numbers less than 6. Note this is the same set as {1,2,3,4,5}. {x | x is a fraction whose numerator is 1 and whose denominator is a counting number }.
August 2006 9 Definition • The set with no elements is called the empty set or the null set and is designated with the symbol ∅.
August 2006 10 Examples – empty set – The set of all pencils in your briefcase might indeed be the empty set. – The set of even prime numbers greater than 2 is the empty set. – The set {x | x < 3 and x > 5} is the empty set.
August 2006 11 Definition - subset • The set A is a subset of the set B if every element of A is an element of B. • If A is a subset of B and B contains elements which are not in A, then A is a proper subset of B.
August 2006 12 Notation - subset If A is a subset of B we write A ⊆ B to designate that relationship. If A is a proper subset of B we write A ⊂ B to designate that relationship. If A is not a subset of B we write A ⊄ B to designate that relationship.
August 2006 13 Example - subset The set A = {1, 2, 3} is a subset of the set B ={1, 2, 3, 4, 5, 6} because each element of A is an element of B. We write A ⊆ B to designate this relationship between A and B. We could also write {1, 2, 3} ⊆ {1, 2, 3, 4, 5, 6}
August 2006 14 Example - subset The set A = {3, 5, 7} is not a subset of the set B = {1, 4, 5, 7, 9} because 3 is an element of A but is not an element of B. The empty set is a subset of every set, because every element of the empty set is an element of every other set.
August 2006 15 PROPER SUBSET: • If A B and A≠B then A is called a⊆ proper subset of B, written as A B.⊂ • For e.g.A={1,3,5},B={1,2,3,4,5,6}
August 2006 16 UNIVERSAL SET: • A set U is called a universal set. If all the sets under consideration are sub-sets of the set U. • For e.g. If A {1,2,3,4},B {2,3,5,7} and C={2,4,6,8}then the universal set U={1,2,3,4,5,6,7,8,9}. POWER SET: • The collection of all possible subsets of a given set A is the power set of A. It is denoted by P(A). • For e.g. If A={1,2,3} then • P(A)={Ø,{1},{2},{3},{1,2},{2,3},{1,3}, {1,2,3}}.
August 2006 17 EQUAL SETS • Two sets A and B are equal if A ⊆ B and B ⊆ A. If two sets A and B are equal we write A = B to designate that relationship.
August 2006 18 Example The sets A = {3, 4, 6} and B = {6, 3, 4} are equal because A ⊆ B and B ⊆ A. The definition of equality of sets shows that the order in which elements are written does not affect the set.
August 2006 19 Example If A = {1, 2, 3, 4, 5} and B = {x | x < 6 and x is a counting number} then A is a subset of B because every element of A is an element of B and B is a subset of A because every element of B is an element of A. Therefore the two sets are equal and we write A = B.
August 2006 20
August 2006 21 Definition - union • The union of two sets A and B is the set containing those elements which are elements of A or elements of B. We write A ∪ B If A = {3, 4, 6} and B = { 1, 2, 3, 5, 6} then A ∪ B = {1, 2, 3, 4, 5, 6}.
August 2006 22 Example - Union If A is the set of prime numbers and B is the set of even numbers then A ∪ B = {x | x is even or x is prime }. If A = {x | x > 5 } and B = {x | x < 3 } then A ∪ B = {x | x < 3 or x > 5 }.
August 2006 23 Venn Diagram - union A is represented by the red circle and B is represented by the blue circle. The purple colored region illustrates the intersection. The union consists of all points which are colored red or blue or purple. A ∪ B A∩B
August 2006 24 Definition - intersection • The intersection of two sets A and B is the set containing those elements which are elements of A and elements of B. We write A ∩ B
August 2006 25 Example If A = {3, 4, 6, 8} and B = { 1, 2, 3, 5, 6} then A ∩ B = {3, 6} If A is the set of prime numbers and B is the set of even numbers then A ∩ B = { 2 } If A = {x | x > 5 } and B = {x | x < 3 } then A ∩ B = ∅
August 2006 26 Example If A = {x | x < 4 } and B = {x | x >1 } then A ∩ B = {x | 1 < x < 4 } If A = {x | x > 4 } and B = {x | x >7 } then A ∩ B = {x | x < 7 }
August 2006 27 These sets can be visualized with circles in what is called a Venn Diagram. A B A ∩ B Everything that is in A AND B.
August 2006 28 • Disjoint sets:Two sets A and B are disjoint sets, if A B=Ø.For e.g.∩ If A={2,4,6,8} and B={ 1,3,5,7} then A and B are disjoint sets. Note: If A B ≠Ø, then A and B are said to∩ be intersecting sets or overlapping sets. • Difference of sets: If A and B are two sets, then their difference A-B is the set containing exactly those element in A that are not in B. Eg:- A={a,b,c} B={a,d} A-B={b,c}
August 2006 29 • Symmetric difference:-Symmetric difference of two sets P&Q is the set containing exactly all the elements that are not in P or in Q but not in both. P(+)Q=(PUQ)-(P Q∩ ) Eg:-P={a,b} Q={a,c} Then symmetric difference is {b,c}
August 2006 30
August 2006 31 • Union and intersection are commutative operations. A ∪ B = B ∪ A A B = B A∩ ∩ • Union and intersection are associative operations. (A ∪ B) ∪ C = A ∪ (B ∪ C) (A B) C = B (A C)∩ ∩ ∩ ∩
August 2006 32 • Idempotent law:- AUA=A A A=A∩ • Identity law:- AU∅=A AUU=U A U=A∩ • Complement law:- AUA’=U U’= ∅
August 2006 33 • Two distributive laws are true. A (∩ B ∪ C )= (A ∩ B) ∪ (A ∩ C) A ∪ ( B ∩ C )= (A ∪ B) ∩ (A ∪ C)

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Introduction to sets

  • 2. August 2006 2 Definitions • A set is a collection of objects. • Objects in the collection are called elements of the set.
  • 3. August 2006 3 Examples - set The collection of persons living in INDIA is a set. – Each person living in INDIA is an element of the set. The collection of all countries in the state of Texas is a set. – Each county in Texas is an element of the set.
  • 4. August 2006 4 Examples - set The collection of counting numbers is a set. – Each counting number is an element of the set. The collection of pencils in your briefcase is a set. – Each pencil in your briefcase is an element of the set.
  • 5. August 2006 5 Notation • Sets are usually designated with capital letters. • Elements of a set are usually designated with lower case letters.
  • 6. August 2006 6 Notation • The roster method of specifying a set consists of surrounding the collection of elements with braces. For example the set of counting numbers from 1 to 5 would be written as {1, 2, 3, 4, 5}.
  • 7. August 2006 7 Notation • Set builder notation has the general form {variable | descriptive statement }. The vertical bar (in set builder notation) is always read as “such that”. Set builder notation is frequently used when the roster method is either inappropriate or inadequate.
  • 8. August 2006 8 Example – set builder notation {x | x < 6 and x is a counting number} is the set of all counting numbers less than 6. Note this is the same set as {1,2,3,4,5}. {x | x is a fraction whose numerator is 1 and whose denominator is a counting number }.
  • 9. August 2006 9 Definition • The set with no elements is called the empty set or the null set and is designated with the symbol ∅.
  • 10. August 2006 10 Examples – empty set – The set of all pencils in your briefcase might indeed be the empty set. – The set of even prime numbers greater than 2 is the empty set. – The set {x | x < 3 and x > 5} is the empty set.
  • 11. August 2006 11 Definition - subset • The set A is a subset of the set B if every element of A is an element of B. • If A is a subset of B and B contains elements which are not in A, then A is a proper subset of B.
  • 12. August 2006 12 Notation - subset If A is a subset of B we write A ⊆ B to designate that relationship. If A is a proper subset of B we write A ⊂ B to designate that relationship. If A is not a subset of B we write A ⊄ B to designate that relationship.
  • 13. August 2006 13 Example - subset The set A = {1, 2, 3} is a subset of the set B ={1, 2, 3, 4, 5, 6} because each element of A is an element of B. We write A ⊆ B to designate this relationship between A and B. We could also write {1, 2, 3} ⊆ {1, 2, 3, 4, 5, 6}
  • 14. August 2006 14 Example - subset The set A = {3, 5, 7} is not a subset of the set B = {1, 4, 5, 7, 9} because 3 is an element of A but is not an element of B. The empty set is a subset of every set, because every element of the empty set is an element of every other set.
  • 15. August 2006 15 PROPER SUBSET: • If A B and A≠B then A is called a⊆ proper subset of B, written as A B.⊂ • For e.g.A={1,3,5},B={1,2,3,4,5,6}
  • 16. August 2006 16 UNIVERSAL SET: • A set U is called a universal set. If all the sets under consideration are sub-sets of the set U. • For e.g. If A {1,2,3,4},B {2,3,5,7} and C={2,4,6,8}then the universal set U={1,2,3,4,5,6,7,8,9}. POWER SET: • The collection of all possible subsets of a given set A is the power set of A. It is denoted by P(A). • For e.g. If A={1,2,3} then • P(A)={Ø,{1},{2},{3},{1,2},{2,3},{1,3}, {1,2,3}}.
  • 17. August 2006 17 EQUAL SETS • Two sets A and B are equal if A ⊆ B and B ⊆ A. If two sets A and B are equal we write A = B to designate that relationship.
  • 18. August 2006 18 Example The sets A = {3, 4, 6} and B = {6, 3, 4} are equal because A ⊆ B and B ⊆ A. The definition of equality of sets shows that the order in which elements are written does not affect the set.
  • 19. August 2006 19 Example If A = {1, 2, 3, 4, 5} and B = {x | x < 6 and x is a counting number} then A is a subset of B because every element of A is an element of B and B is a subset of A because every element of B is an element of A. Therefore the two sets are equal and we write A = B.
  • 21. August 2006 21 Definition - union • The union of two sets A and B is the set containing those elements which are elements of A or elements of B. We write A ∪ B If A = {3, 4, 6} and B = { 1, 2, 3, 5, 6} then A ∪ B = {1, 2, 3, 4, 5, 6}.
  • 22. August 2006 22 Example - Union If A is the set of prime numbers and B is the set of even numbers then A ∪ B = {x | x is even or x is prime }. If A = {x | x > 5 } and B = {x | x < 3 } then A ∪ B = {x | x < 3 or x > 5 }.
  • 23. August 2006 23 Venn Diagram - union A is represented by the red circle and B is represented by the blue circle. The purple colored region illustrates the intersection. The union consists of all points which are colored red or blue or purple. A ∪ B A∩B
  • 24. August 2006 24 Definition - intersection • The intersection of two sets A and B is the set containing those elements which are elements of A and elements of B. We write A ∩ B
  • 25. August 2006 25 Example If A = {3, 4, 6, 8} and B = { 1, 2, 3, 5, 6} then A ∩ B = {3, 6} If A is the set of prime numbers and B is the set of even numbers then A ∩ B = { 2 } If A = {x | x > 5 } and B = {x | x < 3 } then A ∩ B = ∅
  • 26. August 2006 26 Example If A = {x | x < 4 } and B = {x | x >1 } then A ∩ B = {x | 1 < x < 4 } If A = {x | x > 4 } and B = {x | x >7 } then A ∩ B = {x | x < 7 }
  • 27. August 2006 27 These sets can be visualized with circles in what is called a Venn Diagram. A B A ∩ B Everything that is in A AND B.
  • 28. August 2006 28 • Disjoint sets:Two sets A and B are disjoint sets, if A B=Ø.For e.g.∩ If A={2,4,6,8} and B={ 1,3,5,7} then A and B are disjoint sets. Note: If A B ≠Ø, then A and B are said to∩ be intersecting sets or overlapping sets. • Difference of sets: If A and B are two sets, then their difference A-B is the set containing exactly those element in A that are not in B. Eg:- A={a,b,c} B={a,d} A-B={b,c}
  • 29. August 2006 29 • Symmetric difference:-Symmetric difference of two sets P&Q is the set containing exactly all the elements that are not in P or in Q but not in both. P(+)Q=(PUQ)-(P Q∩ ) Eg:-P={a,b} Q={a,c} Then symmetric difference is {b,c}
  • 31. August 2006 31 • Union and intersection are commutative operations. A ∪ B = B ∪ A A B = B A∩ ∩ • Union and intersection are associative operations. (A ∪ B) ∪ C = A ∪ (B ∪ C) (A B) C = B (A C)∩ ∩ ∩ ∩
  • 32. August 2006 32 • Idempotent law:- AUA=A A A=A∩ • Identity law:- AU∅=A AUU=U A U=A∩ • Complement law:- AUA’=U U’= ∅
  • 33. August 2006 33 • Two distributive laws are true. A (∩ B ∪ C )= (A ∩ B) ∪ (A ∩ C) A ∪ ( B ∩ C )= (A ∪ B) ∩ (A ∪ C)