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Set and Set operations, UITM KPPIM DUNGUN

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This document defines sets and common set operations such as union, intersection, difference, complement, Cartesian product, and cardinality. It begins by defining a set as a collection of distinct objects and provides examples of sets. It then discusses ways to represent and visualize sets using listings, set-builder notation, Venn diagrams, and properties of subsets, supersets, equal sets, disjoint sets, and infinite sets. The document concludes by defining common set operations and identities using membership tables and examples.

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SETS and SET OPERATIONS
SET  Definition: A set is a (unordered) collection of objects. These objects are sometimes called elements or members of the set. (Cantor's naive definition) Examples:  Vowels in the English alphabet V = { a, e, i, o, u }  First seven prime numbers. X = { 2, 3, 5, 7, 11, 13, 17 }
SET Representing a set: 1) Listing the members. 2) Definition by property, using set builder notation {x| x has property P} 3) Venn Diagram Example: Even integers between 50 and 63. 1) E = {50, 52, 54, 56, 58, 60, 62} 2) E = {x | 50 <= x < 63, x is an even integer} If enumeration of the members is hard we often use ellipses. Example: a set of positive integers between less than 100 1) A= {1,2,3 …, 99} 2) A = {x ∈ Z+ | x is odd and x < 100}
Sets in Discrete Math  Natural numbers: N = {0,1,2,3, …}  Integers Z = {…, -2,-1,0,1,2, …}  Positive integers Z+ = {1,2, 3.…}  Rational numbers Q = {p/q | p ∈ Z, q ∈ Z, q ≠ 0, the set of rational numbers}  Real numbers R
Special Sets  The universal set is denoted by U: the set of all objects under the consideration.  The empty set is denoted as ∅ or { }.
Venn Diagram  A set can be visualized using Venn Diagrams: V={ A, B, C } A B C U
Comparing Sets Subsets Superset Equal sets Disjoint sets Infinite sets
Subset  Definition: A set A is said to be a subset of B if and only if every element of A is also an element of B. We use A ⊆ B to indicate A is a subset of B.  Alternate way to define A is a subset of B: ∀x (x ∈ A) → (x ∈ B) A B
Subset Properties  Theorem ∅ ⊆ S Empty set is a subset of any set.  Theorem: S ⊆ S Any set S is a subset of itself
Superset  Definition: A set B is a superset of another set A if all elements of the set A are elements of the set B.  A⊃B
Equal Sets  Definition: Two sets are equal if and only if they have the same elements. Example:  {1,2,3} = {3,1,2} = {1,2,1,3,2} – listed more than one, they have the same elements Note: Duplicates don't contribute anything new to a set, so remove them. The order of the elements in a set doesn't contribute anything new.
Disjoint Sets  Two sets are called disjoint if their intersection is the empty set. Example A = {1, 3, 5, 7, 9} and B = {2, 4, 6, 8, 10}. A ∩ B = ∅, A and B are disjoint A B U
Infinite Set  Definition: A set is infinite if it is not finite. Examples:  The set of natural numbers is an infinite set. N = {0, 1, 2, 3, ... }  The set of real numbers is an infinite set.
Set Operators  Union, ∪  Intersection, ∩  Complement,  Difference, -  Proper Subset, ⊂  Powerset, P  Cardinality, |..|  Cartesian Product, x
Union  Definition: Let A and B be sets. The union of 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.  Alternate: A ∪ B = { x | x ∈ A ∨ x ∈ B } Example: A = {1,2,3,6} B = { 2,4,6,9} A ∪ B = { 1,2,3,4,6,9 }
Intersection  Definition: Let A and B be sets. The intersection of A and B, denoted by A ∩ B, is the set that contains those elements that are in both A and B.  Alternate: A ∩ B = { x | x ∈ A ∧ x ∈ B } Example: A = {1,2,3,6} B = { 2, 4, 6, 9} A ∩B = { 2, 6 }
Complement  Let U be the universal set. The complement of the set A, denoted by A, is the complement of A with respect to U. Therefore, the complement of the set A is U − A. A = {x ∈ U | x /∈ A}. Example A = {a, e, i, o, u} (where the universal set is the set of letters of the English alphabet) A = {b, c, d, f, g, h, j, k, l,m, n, p, q, r, s, t, v,w, x, y, z}.
Difference  Definition: Let A and B be sets. The difference of A and B, denoted by A - B, is the set containing those elements that are in A but not in B. The difference of A and B is also called the complement of B with respect to A.  A - B = { x | x ∈ A ∧ x ∉ B }. Example: A= {1,2,3,5,7} B = {1,5,6,8} A - B ={2,3,7}
A Proper Subset  Definition: A set A is said to be a proper subset of B if and only if A ⊆ B and A ≠ B. We denote that A is a proper subset of B with the notation A ⊂ B. A B
A Proper Subset Example: A={1,2,3} B ={1,2,3,4,5} Is: A ⊂ B ? Yes.
Power Set  Definition: Given a set S, the power set of S is the set of all subsets of S. The power set is denoted by P(S). Examples: Assume an empty set ∅  What is the power set of ∅ ? P(∅) = { ∅ }  What is the cardinality of P(∅) ? |P(∅)| = 1 Assume set {1}  P( {1} ) = { ∅, {1} }  |P({1})| = 2
Power Set Assume {1,2}  P( {1,2} ) = { ∅, {1}, {2}, {1,2} }  |P({1,2} )| =4 Assume {1,2,3}  P({1,2,3}) = {∅, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} }  |P({1,2,3} | = 8  If S is a set with |S| = n then | P(S) | = 2n..
Cardinality  Definition: Let S be a set. If there are exactly n distinct elements in S, where n is a nonnegative integer, we say S is a finite set and that n is the cardinality of S. The cardinality of S is denoted by | S |. Examples:  V={1 2 3 4 5} | V | = 5  A={1,2,3,4, …, 20} |A| =20  | ∅ | = 0
Cartesian Product  Definition: Let S and T be sets. The Cartesian product of S and T, denoted by S x T, is the set of all ordered pairs (s,t), where s∈ S and t ∈ T. Hence, S x T = { (s,t) | s ∈ S ∧ t ∈ T}. Examples:  S = {1,2} and T = {a,b,c}  S x T = { (1,a), (1,b), (1,c), (2,a), (2,b), (2,c) }  T x S = { (a,1), (a, 2), (b,1), (b,2), (c,1), (c,2) }  Note: S x T ≠ T x S !!!!
Cardinality of Cartesian Product  |S x T| = |S| * |T| Example:  A= {John, Peter, Mike}  B ={Jane, Ann, Laura}  A x B= {(John, Jane), (John, Ann) , (John, Laura), (Peter, Jane), (Peter, Ann) , (Peter, Laura) , (Mike, Jane) , (Mike, Ann) , (Mike, Laura)}  |A x B| = 9  |A|=3, |B|=3 -> |A||B|= 9 Definition: A subset of the Cartesian product A x B is called a relation from the set A to the set B.
Set Identities
Set Identities  Set identities can be proved using membership tables.  List each combination of sets that an element can belong to.  Then show that for each such a combination the element either belongs or does not belong to both sets in the identity. Example Prove: (A ∩ B) = A ∪ B A B A B A ∩ B A ∪ B 1 1 0 0 0 0 1 0 0 1 1 1 0 1 1 0 1 1 0 0 1 1 1 1

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Set and Set operations, UITM KPPIM DUNGUN

  • 2. SET  Definition: A set is a (unordered) collection of objects. These objects are sometimes called elements or members of the set. (Cantor's naive definition) Examples:  Vowels in the English alphabet V = { a, e, i, o, u }  First seven prime numbers. X = { 2, 3, 5, 7, 11, 13, 17 }
  • 3. SET Representing a set: 1) Listing the members. 2) Definition by property, using set builder notation {x| x has property P} 3) Venn Diagram Example: Even integers between 50 and 63. 1) E = {50, 52, 54, 56, 58, 60, 62} 2) E = {x | 50 <= x < 63, x is an even integer} If enumeration of the members is hard we often use ellipses. Example: a set of positive integers between less than 100 1) A= {1,2,3 …, 99} 2) A = {x ∈ Z+ | x is odd and x < 100}
  • 4. Sets in Discrete Math  Natural numbers: N = {0,1,2,3, …}  Integers Z = {…, -2,-1,0,1,2, …}  Positive integers Z+ = {1,2, 3.…}  Rational numbers Q = {p/q | p ∈ Z, q ∈ Z, q ≠ 0, the set of rational numbers}  Real numbers R
  • 5. Special Sets  The universal set is denoted by U: the set of all objects under the consideration.  The empty set is denoted as ∅ or { }.
  • 6. Venn Diagram  A set can be visualized using Venn Diagrams: V={ A, B, C } A B C U
  • 8. Subset  Definition: A set A is said to be a subset of B if and only if every element of A is also an element of B. We use A ⊆ B to indicate A is a subset of B.  Alternate way to define A is a subset of B: ∀x (x ∈ A) → (x ∈ B) A B
  • 9. Subset Properties  Theorem ∅ ⊆ S Empty set is a subset of any set.  Theorem: S ⊆ S Any set S is a subset of itself
  • 10. Superset  Definition: A set B is a superset of another set A if all elements of the set A are elements of the set B.  A⊃B
  • 11. Equal Sets  Definition: Two sets are equal if and only if they have the same elements. Example:  {1,2,3} = {3,1,2} = {1,2,1,3,2} – listed more than one, they have the same elements Note: Duplicates don't contribute anything new to a set, so remove them. The order of the elements in a set doesn't contribute anything new.
  • 12. Disjoint Sets  Two sets are called disjoint if their intersection is the empty set. Example A = {1, 3, 5, 7, 9} and B = {2, 4, 6, 8, 10}. A ∩ B = ∅, A and B are disjoint A B U
  • 13. Infinite Set  Definition: A set is infinite if it is not finite. Examples:  The set of natural numbers is an infinite set. N = {0, 1, 2, 3, ... }  The set of real numbers is an infinite set.
  • 14. Set Operators  Union, ∪  Intersection, ∩  Complement,  Difference, -  Proper Subset, ⊂  Powerset, P  Cardinality, |..|  Cartesian Product, x
  • 15. Union  Definition: Let A and B be sets. The union of 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.  Alternate: A ∪ B = { x | x ∈ A ∨ x ∈ B } Example: A = {1,2,3,6} B = { 2,4,6,9} A ∪ B = { 1,2,3,4,6,9 }
  • 16. Intersection  Definition: Let A and B be sets. The intersection of A and B, denoted by A ∩ B, is the set that contains those elements that are in both A and B.  Alternate: A ∩ B = { x | x ∈ A ∧ x ∈ B } Example: A = {1,2,3,6} B = { 2, 4, 6, 9} A ∩B = { 2, 6 }
  • 17. Complement  Let U be the universal set. The complement of the set A, denoted by A, is the complement of A with respect to U. Therefore, the complement of the set A is U − A. A = {x ∈ U | x /∈ A}. Example A = {a, e, i, o, u} (where the universal set is the set of letters of the English alphabet) A = {b, c, d, f, g, h, j, k, l,m, n, p, q, r, s, t, v,w, x, y, z}.
  • 18. Difference  Definition: Let A and B be sets. The difference of A and B, denoted by A - B, is the set containing those elements that are in A but not in B. The difference of A and B is also called the complement of B with respect to A.  A - B = { x | x ∈ A ∧ x ∉ B }. Example: A= {1,2,3,5,7} B = {1,5,6,8} A - B ={2,3,7}
  • 19. A Proper Subset  Definition: A set A is said to be a proper subset of B if and only if A ⊆ B and A ≠ B. We denote that A is a proper subset of B with the notation A ⊂ B. A B
  • 20. A Proper Subset Example: A={1,2,3} B ={1,2,3,4,5} Is: A ⊂ B ? Yes.
  • 21. Power Set  Definition: Given a set S, the power set of S is the set of all subsets of S. The power set is denoted by P(S). Examples: Assume an empty set ∅  What is the power set of ∅ ? P(∅) = { ∅ }  What is the cardinality of P(∅) ? |P(∅)| = 1 Assume set {1}  P( {1} ) = { ∅, {1} }  |P({1})| = 2
  • 22. Power Set Assume {1,2}  P( {1,2} ) = { ∅, {1}, {2}, {1,2} }  |P({1,2} )| =4 Assume {1,2,3}  P({1,2,3}) = {∅, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} }  |P({1,2,3} | = 8  If S is a set with |S| = n then | P(S) | = 2n..
  • 23. Cardinality  Definition: Let S be a set. If there are exactly n distinct elements in S, where n is a nonnegative integer, we say S is a finite set and that n is the cardinality of S. The cardinality of S is denoted by | S |. Examples:  V={1 2 3 4 5} | V | = 5  A={1,2,3,4, …, 20} |A| =20  | ∅ | = 0
  • 24. Cartesian Product  Definition: Let S and T be sets. The Cartesian product of S and T, denoted by S x T, is the set of all ordered pairs (s,t), where s∈ S and t ∈ T. Hence, S x T = { (s,t) | s ∈ S ∧ t ∈ T}. Examples:  S = {1,2} and T = {a,b,c}  S x T = { (1,a), (1,b), (1,c), (2,a), (2,b), (2,c) }  T x S = { (a,1), (a, 2), (b,1), (b,2), (c,1), (c,2) }  Note: S x T ≠ T x S !!!!
  • 25. Cardinality of Cartesian Product  |S x T| = |S| * |T| Example:  A= {John, Peter, Mike}  B ={Jane, Ann, Laura}  A x B= {(John, Jane), (John, Ann) , (John, Laura), (Peter, Jane), (Peter, Ann) , (Peter, Laura) , (Mike, Jane) , (Mike, Ann) , (Mike, Laura)}  |A x B| = 9  |A|=3, |B|=3 -> |A||B|= 9 Definition: A subset of the Cartesian product A x B is called a relation from the set A to the set B.
  • 27. Set Identities  Set identities can be proved using membership tables.  List each combination of sets that an element can belong to.  Then show that for each such a combination the element either belongs or does not belong to both sets in the identity. Example Prove: (A ∩ B) = A ∪ B A B A B A ∩ B A ∪ B 1 1 0 0 0 0 1 0 0 1 1 1 0 1 1 0 1 1 0 0 1 1 1 1