ELEMENTARY SET THEORY. Sets A set is a well-defined collection of objects. A set is a well-defined collection of objects. Each object in a set is called

ELEMENTARY SET THEORYELEMENTARY SET THEORY

SetsSets

A A setset is a well-defined collection of is a well-defined collection of objects.objects.

Each object in a set is called an Each object in a set is called an element element or or membermember of the set.of the set.

The elements or objects of the set The elements or objects of the set are enclosed by a pair of braces { }. are enclosed by a pair of braces { }.

NotationsNotations

Capital or uppercase letters are Capital or uppercase letters are usually used to denote sets while usually used to denote sets while small or lowercase letters denote small or lowercase letters denote elements of a set.elements of a set.

denotes “is an element of” or denotes “is an element of” or “belongs to”“belongs to”

denotes “is not an element of” or denotes “is not an element of” or “does not belong to”“does not belong to”

ExampleExample

Let A – the set of letters in the Let A – the set of letters in the English alphabetEnglish alphabet

B – the set of primary colors B – the set of primary colors C – the set of positive integers C – the set of positive integers

g g A Aorange orange B B100 100 C C

Ways of Describing a SetWays of Describing a Set

List (or roster) methodList (or roster) method describes a set describes a set by enumerating the elements of the set.by enumerating the elements of the set.

A = {a, b, c, d,…,z}A = {a, b, c, d,…,z}B = {red, yellow, blue}B = {red, yellow, blue}C = {1, 2, 3, 4,…}C = {1, 2, 3, 4,…}

Rule (or set builder) methodRule (or set builder) method describes describes a set by a statement or a rule.a set by a statement or a rule.

A = {x|x is an English alphabet}A = {x|x is an English alphabet}B = {a|a is a primary color}B = {a|a is a primary color}C = {y|y is a positive integer}C = {y|y is a positive integer}

Definition of TermsDefinition of Terms

The The cardinalitycardinality of a set A, or the cardinal of a set A, or the cardinal number of A, denoted as n(A), is the number of A, denoted as n(A), is the number of elements in A.number of elements in A.

n(A) = 26, n(B) = 3, n(C) = n(A) = 26, n(B) = 3, n(C) = A set is A set is finitefinite if there is one counting if there is one counting

number that indicates the total number of number that indicates the total number of elements in the set.elements in the set.

A and B are finite sets.A and B are finite sets. A set is A set is infiniteinfinite if in counting the if in counting the

elements, we never come to an end.elements, we never come to an end.C is an infinite set.C is an infinite set.


The The null setnull set or or empty setempty set, denoted by the , denoted by the symbol symbol , is the set that contains no , is the set that contains no elements, that is, A is empty iff n(A) = 0.elements, that is, A is empty iff n(A) = 0.

D = {x|x is a month in the Gregorian D = {x|x is a month in the Gregorian calendar with less than 28 days}calendar with less than 28 days}

n(D) = 0, so D = n(D) = 0, so D = A A singleton setsingleton set is a set that contains only is a set that contains only

one element, that is, B is a singleton set iff one element, that is, B is a singleton set iff n(B) = 1.n(B) = 1.

E = {y|y is prime number, 5 E = {y|y is prime number, 5 < y < 10}< y < 10}


Sets A and B are Sets A and B are equalequal if they have the if they have the same elements.same elements.

Set A is a Set A is a subsetsubset of B, denoted as A of B, denoted as A B, B, iff every element of A is also an element iff every element of A is also an element of B.of B.Laws of subset:Laws of subset: Every set is a subset of itself, that is, A Every set is a subset of itself, that is, A A, A,

for any set A.for any set A. The null set is a subset of any set, that is, The null set is a subset of any set, that is,

A, for any set A. A, for any set A.


Example: SubsetExample: SubsetG = {x|x is an integer}G = {x|x is an integer}F = {y|y is a whole number}F = {y|y is a whole number}C = {z|z is a positive integer}C = {z|z is a positive integer}

C C G because every element of C is G because every element of C is found in G.found in G.F F C because 0 C because 0F but 0F but 0C.C.


Set A is a Set A is a proper subsetproper subset of B, denoted of B, denoted AAB, if A is a subset of B and there is at B, if A is a subset of B and there is at least one element of B that is not in A. least one element of B that is not in A. That is, AThat is, AB iff AB iff AB and AB and AB.B.

P = {1, 3, 5, 7}P = {1, 3, 5, 7}Q = {3, 7}Q = {3, 7}Then, QThen, QPP

The set containing all of the elements for The set containing all of the elements for any particular discussion is called the any particular discussion is called the universal setuniversal set, denoted as U. , denoted as U.

ExerciseExercise

Describe the following sets using the list Describe the following sets using the list method and give the set cardinality:method and give the set cardinality:

a.a. A = {x|x is a natural number which is 1 less A = {x|x is a natural number which is 1 less than a multiple of 3}than a multiple of 3}

b.b. B = {a|a is a rational number whose value is B = {a|a is a rational number whose value is 2/3}2/3}

c.c. C = {b|b is a vowel that appears in the C = {b|b is a vowel that appears in the phrase “set of vowels”}phrase “set of vowels”}

d.d. D = {z|z is an even prime integer greater D = {z|z is an even prime integer greater than 2}than 2}

ExerciseExercise

Describe the following sets using Describe the following sets using the rule method:the rule method:

a.a. F = {0, 1, 8, 27, 64, 125, 216, …}F = {0, 1, 8, 27, 64, 125, 216, …}

b.b. G = {…, -30, -20, -10, 0, 10, 20, 30, G = {…, -30, -20, -10, 0, 10, 20, 30, 40, …}40, …}

c.c. H = {1, 2, 3, 5, 7, 11, 13, 17, 19, 23, H = {1, 2, 3, 5, 7, 11, 13, 17, 19, 23, …}…}

ExerciseExercise

Determine which of the following Determine which of the following statements are true and which are false:statements are true and which are false:

a.a. NNb.b. {1, 2, 3} {1, 2, 3} N Nc.c. {0} {0} N Nd.d. {1, 2, 3} {1, 2, 3} {1, 2, 3} {1, 2, 3}e.e. 1 1 {1, 2, 3} {1, 2, 3}f.f. n(N) = 10n(N) = 101010

g.g. A A B B n(A) n(A) > n(B)> n(B)h.h. {3} {3} N N

The Power SetThe Power Set

Given a set S, the Given a set S, the power setpower set of S is the of S is the set of all subsets of the set S. The set of all subsets of the set S. The power set of S is denoted by P(S) or power set of S is denoted by P(S) or (S).(S).

Example: Example: S = {0, 1, 2} S = {0, 1, 2}

(S) = {(S) = {, {0}, {1}, {2}, {0, 1}, {0, , {0}, {1}, {2}, {0, 1}, {0, 2}, {1, 2}, {0, 1, 2}}2}, {1, 2}, {0, 1, 2}} Note that the empty set and the set itself Note that the empty set and the set itself

are members of this set of subsets.are members of this set of subsets.

The Power SetThe Power Set

If a set has If a set has nn elements, then its power elements, then its power set has 2set has 2nn elements. elements.

Exercise:Exercise: What is the power set of A = {x, y, z}?What is the power set of A = {x, y, z}? What is the power set of the null set?What is the power set of the null set? What is the power set of the power set of What is the power set of the power set of

the null set?the null set? What is the power set of B = {0, {1}, 3, What is the power set of B = {0, {1}, 3,

{2, 4}}{2, 4}}

SET OPERATIONSSET OPERATIONS

UnionUnion

Let A and B be sets. The Let A and B be sets. The unionunion of the of the sets A and B, denoted by A sets A and B, denoted by A B, is B, is the set that contains those elements the set that contains those elements that are either in A or in B, or in both.that are either in A or in B, or in both.

A A B = {x|x B = {x|x A A x x B} B} Example:Example:

A = {1, 3, 5} and B = {1, 2, 3}A = {1, 3, 5} and B = {1, 2, 3}A A B = {1, 2, 3, 5} B = {1, 2, 3, 5}

UnionUnion

Venn DiagramVenn Diagram

IntersectionIntersection

Let A and B be sets. The Let A and B be sets. The intersectionintersection of the sets A and B, denoted by A of the sets A and B, denoted by A B, is the set containing those B, is the set containing those elements in both A and B.elements in both A and B.

A A B = {x|x B = {x|x A A x x B} B} Example: Example:

A = {1, 3, 5} and B = {1, 2, 3}A = {1, 3, 5} and B = {1, 2, 3}A A B = {1, 3} B = {1, 3}

IntersectionIntersection


DisjointDisjoint

Two sets are called Two sets are called disjointdisjoint if their if their intersection is the empty set.intersection is the empty set.

A A B = { } B = { } Example: Example:

A = {1, 3, 5, 7, 9}A = {1, 3, 5, 7, 9}B = {2, 4, 6, 8, 10}B = {2, 4, 6, 8, 10}A A B = B = , then A and B are , then A and B are

disjointdisjoint

DifferenceDifference

Let A and B be sets. The Let A and B be sets. The differencedifference of A of A and B, denoted by A – B, is the set and B, denoted by A – B, is the set containing those elements that are in A containing those elements that are in A but not in B.but not in B.

The difference of A and B is also called the The difference of A and B is also called the complement of B with respect to A.complement of B with respect to A.

A – B = {x|x A – B = {x|x A A x x B} B} Example:Example:

A = {1, 3, 5} and B = {1, 2, 3}A = {1, 3, 5} and B = {1, 2, 3}A – B = {5}A – B = {5}

DifferenceDifference


ComplementComplement

Let Let UU be the universal set. The be the universal set. The complementcomplement of the set A, denoted by A’, of the set A, denoted by A’, is the complement of A with respect to is the complement of A with respect to UU. . In other words, the complement of set A In other words, the complement of set A is is UU – A. – A.

A’ = {x|x A’ = {x|x A} A} Example:Example:

A = {a, e, i, o, u}A = {a, e, i, o, u}where the universal set is the set of where the universal set is the set of

letters in the English alphabetletters in the English alphabetA’ = {y|y is a consonant}A’ = {y|y is a consonant}

ComplementComplement


Cartesian ProductsCartesian Products

The order of elements in a collection is often The order of elements in a collection is often important. Since sets are unordered, a different important. Since sets are unordered, a different structure is needed to represent ordered structure is needed to represent ordered collections. This is provided by collections. This is provided by ordered n-tuplesordered n-tuples..

The The ordered n-tupleordered n-tuple (a (a11, a, a22,…a,…ann) is the ordered ) is the ordered collection that has acollection that has a11 as its first element, a as its first element, a22 as its as its second element,…, and asecond element,…, and an n as its as its nnth element.th element.

Two ordered Two ordered nn-tuples are equal iff each -tuples are equal iff each corresponding pair of their elements is equal.corresponding pair of their elements is equal.


Let A and B be sets. The Let A and B be sets. The Cartesian Cartesian productproduct of A and B, denoted by A x B, of A and B, denoted by A x B, is the set of all ordered pairs (a, b) is the set of all ordered pairs (a, b) where a where a A and b A and b B. B.

A x B = {(a, b)|a A x B = {(a, b)|a A A b b B} B} Example: A = {1, 2} and B = {a, b, c}Example: A = {1, 2} and B = {a, b, c}

A x B = {(1, a), (1, b), (1, c), (2, A x B = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)}a), (2, b), (2, c)}

Note: A x B Note: A x B B x A B x A


The The Cartesian productCartesian product of sets A of sets A11, A, A22,…, ,…, AAnn, denoted by A, denoted by A11 x A x A22 x … x A x … x Ann is the is the set of ordered set of ordered nn-tuples (a-tuples (a11, a, a22,…, a,…, ann), ), where awhere aii belongs to A belongs to Aii for i = 1, 2,…, n. for i = 1, 2,…, n.

AA11 x A x A22 x … x A x … x Ann = {(a = {(a11, a, a22, …, a, …, ann) | a) | aii AAii for i = 1, 2, …, n} for i = 1, 2, …, n}

Example: A = {0, 1}, B = {1, 2}, Example: A = {0, 1}, B = {1, 2}, C = {0, 1, 2}, find A x B x C. C = {0, 1, 2}, find A x B x C.

ExerciseExercise

Let Let AA be the set of students who live be the set of students who live one km from school and let one km from school and let BB be the be the set of students who walk to classes. set of students who walk to classes. Describe the students in each of these Describe the students in each of these sets.sets.

a.a. A A B B

b.b. A A B B

c.c. A – BA – B

d.d. B – AB – A

ExerciseExercise

Let A = {1, 2, 3, 4, 5} and B = {0, 3, 6}. Let A = {1, 2, 3, 4, 5} and B = {0, 3, 6}. FindFind

a.a. A A B B

b.b. A A B B

c.c. A – BA – B

d.d. B – A B – A Let A = {0, 2, 4, 6, 8, 19}, B = {0, 1, 2, Let A = {0, 2, 4, 6, 8, 19}, B = {0, 1, 2,

3, 4, 5, 6} and C = {4, 5, 6, 7, 8, 9, 10}3, 4, 5, 6} and C = {4, 5, 6, 7, 8, 9, 10}

a. A a. A B B C C b. A b. A B B C C

c. (A c. (A B) B) C C d. (A d. (A B) B) C C

Properties of Set UnionProperties of Set Union

For any sets A, B, and C,For any sets A, B, and C,1.1. The union of any set with the null set is The union of any set with the null set is

the set itself. A the set itself. A = A = A2.2. The union of any set with itself is the set The union of any set with itself is the set

itself. A itself. A A = A A = A3.3. Set Union is commutative. A Set Union is commutative. A B = B B = B A A4.4. Set Union is associative. (A Set Union is associative. (A B) B) C = A C = A

(B (B C) C)5.5. Any set is a subset of its union with Any set is a subset of its union with

another set. A another set. A A A B B

Properties of Set Properties of Set IntersectionIntersection

For any sets A, B, and C,For any sets A, B, and C,1.1. The intersection of any set with the null set is The intersection of any set with the null set is

the null set. A the null set. A = = 2.2. The intersection of any set with itself is the set The intersection of any set with itself is the set

itself. A itself. A A = A A = A3.3. Set Intersection is commutative. A Set Intersection is commutative. A B = B B = B A A4.4. Set Intersection is associative. (A Set Intersection is associative. (A B) B) C = A C = A

(B (B C) C)5.5. The intersection of any given set with another The intersection of any given set with another

set is a subset of the given set. A set is a subset of the given set. A B B A A6.6. A A (B (B C) = (A C) = (A B) B) (A (A C) C)

A A (B (B C) = (A C) = (A B) B) (A (A C) C)

Properties of Set DifferenceProperties of Set Difference

For any sets A, B, and C,For any sets A, B, and C,1.1. The removal of the null set from any set The removal of the null set from any set

has no effect on the set. A – has no effect on the set. A – = A = A2.2. The removal of the elements of any set The removal of the elements of any set

from itself will leave the empty set. A – A from itself will leave the empty set. A – A = =

3.3. No elements can be removed from the No elements can be removed from the null set. null set. – A = – A =

4.4. The result of removing the elements of a The result of removing the elements of a set from any given set is a subset of the set from any given set is a subset of the given set. A – B given set. A – B A A

5.5. A – (B A – (B C) = (A – B) C) = (A – B) (A – C) (A – C)A – (B A – (B C) = (A – B) C) = (A – B) (A – C) (A – C)

Properties of Set CardinalityProperties of Set Cardinality

For any sets A, B, and C,For any sets A, B, and C,

1.1. |A |A B| = |A| – |A – B| B| = |A| – |A – B|

2.2. |A |A B| = |A| + |B| – |A B| = |A| + |B| – |A B| B|

3.3. |A – B| = |A| – |A |A – B| = |A| – |A B| B|

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ELEMENTARY SET THEORY. Sets A set is a well-defined collection of objects. A set is a well-defined collection of objects. Each object in a set is called