Set Theory

  • Set Theory
  • Counting Rules:
  • The Rule of Multiplication

“SET”

statistics and probability  Set Theory

A set is any welldefined collection or list of distinct objects, e.g. a group of students, the books in a library, the integers between 1 and 100, all human beings on the earth, etc. The term welldefined here means that any object must be classified as either belonging or not belonging to the set under consideration, and the term distinct implies that each object must appear only once. The objects that are in a set, are called members or elements of that set. Sets are usually denoted by capital letters such as A, B, C, Y, etc., while their elements are represented by small letters such as, a, b, c, y, etc. Elements are enclosed by parentheses to represent a set. For example:

EXAMPLES OF SETS:

A ={a,b,c, d} or

B ={1,2,3,7} The Number of a set A, written as n(A), is defined as the number of elements in A. If x is an element of a set A, we write x ∈ A which is read as “x belongs to A” or x is in A. If x does not belong to A,

i.e. x is not an element of A, we write x ∉ A. A set that has no elements is called an empty or a null set and is denoted by the symbolφ. (It must be noted that {0} is not an empty set as it contains an element 0.) If a set contains only one element, it is called a unit set or a singleton set. It is also important to note the difference between an element “x” and a unit set {x}. A set may be specified in two ways:

1. We may give a list of all the elements of a set (the “Roster” method),

e.g. A ={1,3,5,7,9, 11}; B = {a book, a city, a clock, a teacher};

2. We may state a rule that enables us to determine whether or not a given object is a member of the set(the “Rule” method or the “Set Builder” method),

e.g.

A = {x | x is an odd number and x < 12} meaning that A is a set of all elements x such that x is an odd number and x is less than 12. (The vertical line is read as “such that”.). An important point to note is that:The repetition or the order in which the elements of a set occur, does not change the nature of the set. The size of a set is given by the number of elements present in it. This number may be finite or infinite. Thus a set is finite when it contains a finite number of elements; otherwise it is an infinite set. The Empty set is regarded as a Finite set.

EXAMPLES OF FINITE SETS

i) A = {1, 2, 3……., 99, 100};

ii) B={x|xisamonthof the year};

iii) C={x|xisaprinting mistake in a book};

iv) D = {x | x is a living citizen of Pakistan}; Examples of infinite sets:

i) A={x|xisaneven integer}; ii) B={x|xisarealnumber between 0 and 1 inclusive},

i.e.B = (x |x 0 < x <1} iii) C={x|xisapointonaline}; iv) D={x|xisasentenceina

English language}; etc

SUBSETS

A set that consists of some elements of another set, is called a subset of that set. For example, if B is a subset of A, then every member of set B is also a member of set A.

If B is a subset of A, we write: B ⊂ A or equivalently: A ⊃ B ‘B is a subset of A’ is also read as ‘B is contained in A’, or ‘A contains B’.

EXAMPLE

If A={1,2,3,4,5,10} and B{1,3,5} then B ⊂ A,

i.e. B is contained in A. It should be noted that any set is always regarded a subset of itself. and an empty set φ is considered to be a subset of every set. Two sets A and B are Equal or Identical, if and only if they contain exactly the same elements. In other words, A = B if and only if A ⊂ B and B ⊂ A.

PROPER SUBSET

If a set B contains some but not all of the elements of another set A, while A contains each element of B, i.e. if B ⊂ A and B ≠ A then the set B is defined to be a proper subset of A.

Universal Set:

The original set of which all the sets we talk about, are subsets, is called the universal set (or the space) and is generally denoted by S or Ω. The universal set thus contains all possible elements under consideration. A set S with n elements will produce 2n subsets, including S and φ.

EXAMPLE;

Consider the set A = {1, 2, 3}. All possible subsets of this set are: φ, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3} and {1, 2, 3} Hence, there are 23 = 8 subsets of the set A.

VENN DIAGRAM

A diagram that is understood to represent sets by circular regions, parts of circular regions or their complements with respect to a rectangle representing the space S is called a Venn diagram, named after the English logician John Venn (18341923). The Venn diagrams are used to represent sets and subsets in a pictorial way and to verify the relationship among sets and subsets.

A Simple Venn diagram:

Disjoint Sets

statistics and probability  Set Theory

statistics and probability  Set Theory

OPERATIONS ON SETS

Let the sets A and B be the subsets of some universal set S. Then these sets may be combined and operated on in various ways to form new sets which are also subsets of S. The basic operations are union, intersection, difference and complementation.

UNION OF SETS

The union or sum of two sets A and B, denoted by A ∪ B, and read as “A union B”, means the set of all elements that belong to at least one of the sets A and B, that is A ∪ B={x|x ∈ A or x ∈ B} By means of a Venn Diagram, A ∪ B is shown by the shaded area as below:

statistics and probability  Set Theory

EXAMPLE

Let A ={1,2,3,4}and B ={3,4, 5,6} Then A ∪ B = {1,2,3,4, 5, 6}

INTERSECTION OF SETS

The intersection of two sets A and B, denoted by A ∩ B, and read as “A intersection B”, means that the set of all elements that belong to both A and B; that is A ∩ B ={x |x ∈ and x ∈ B}. Diagrammatically, A ∩ B is shown by the shaded area as below:

statistics and probability  Set Theory

EXAMPLE

Let A ={1,2,3,4}and B ={3,4, 5,6} Then A ∩ B = {3, 4} The operations of union and intersection that have been defined for two sets may conveniently be extended to any finite number of sets.

DISJOINT SETS

Two sets A and B are defined to be disjoint or mutually exclusive or nonoverlapping when they have no elements in common, i.e. when their intersection is an empty set

i.e. A ∩ B= φ. On the other hand, two sets A and B are said to be conjoint when the have at least one element in common.

SET DIFFERENCE

The difference of two sets A and B, denoted by A – B or by A – (A ∩ B), is the set of all elements of A which do not belong to B. Symbolically, A – B ={x |x ∈ A and x ∉ B}

It is to be pointed out that in general A – B ≠ B – A. The shaded area of the following Venn diagram shows the difference A – B:

statistics and probability  Set Theory

Difference A – B is shaded

It is to be noted that A – B and B are disjoint sets. If A and B are disjoint, then the difference A – B coincides with the set A.

COMPLEMENTATION

The particular difference S – A, that is, the set of all those elements of S which do not belong to A, is called the complement of A and is denoted by A or by Ac. In symbols:

A ={x |x ∈ S and s ∉ A}

statistics and probability  Set Theory

A is shaded

It should be noted that A – B and A ∩ B, where  B is the complement of set B, are the same set. Next, we consider the Algebra of Sets. The algebra of sets provides us with laws which can be used to solve many problems in probability calculations. Let A, B and C be any subsets of the universal set S. Then, we have:

1. Commutative laws:

A ∪ B=B ∪ A A ∩ B=B ∩ A

2. Associative laws:

(A ∪ B) ∪ C=A ∪ (B ∪ C)

(A ∩ B) ∩ C=A ∩ (B ∩ C)

3. Distributive laws

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

4. Idempotent laws

A ∪ A=A A ∩ A=A

5. Identity laws

A ∪ S = S, A ∩ S = A, A ∪φ = A, and A ∩φ = φ.

6. Complementation laws

A ∪ A = S, A  ∩A= φ,

( A) =A,

S= φ, and φ  =S

7. De Morgan’s laws:

(A ∪ B)= A ∩ B, and (A ∩ B)= A ∪ B

PARTITION OF SETS

A partition of a set S is a subdivision of the set into nonempty subsets that are disjoint and exhaustive, i.e. their union is the set S itself. This implies that:

• i)Ai ∩ Aj = φ, where i ≠ j;

• ii)A1 ∩ A2 ∪ … ∪ An =S. The subsets in a partition are called cells.

EXAMPLE

Let us consider a set S = {a, b, c, d, e}. Then {a, b}, and {c, d, e} is a partition of S as each element of S belongs to exactly one cell.

CLASS OF SETS

A set of sets is called a class. For example, in a set of lines, each line is a set of points.

POWER SET

The class of ALL subsets of a set A is called the Power Set of A and is denoted by P(A). For example, if A = {H, T}, then P(A) = {φ, {H}, {T}, {H, T}}.

CARTESIAN PRODUCT OF SETS

The Cartesian product of sets A and B, denoted by A × B, (read as “A cross B”), is a set that contains all ordered pairs (x, y), where x belongs to A and y belongs to B. Symbolically, we write A × B = {(x, y) |x ∈ A and y ∈ B} This set is also called the Cartesian set of A and B set of A and B, named after the French mathematician Rene’ Descartes (15961605). The product of a set A by itself is denoted by A2. This concept of product may be extended to any finite number of sets.

EXAMPLE

Let A = {H, T} and B = {1, 2, 3, 4, 5, 6}. Then the Cartesian product set is the collection of the following twelve (2 × 6) ordered pairs: A×B = {(H, 1); (H, 2);(H, 3); (H, 4);(H, 6); (H, 6);(T, 1); (T, 2); (T, 3); (T, 4); (T, 5); (T, 6) }

statistics and probability  Set Theory

The plural of the word ‘die’ is ‘dice’. The product A × B may conveniently be found by means of the socalled tree diagram shown below:

Tree Diagram

A BA B


statistics and probability  Set Theory statistics and probability  Set Theory

1 1 1 1 1 1

1 2 3 4 5 6 (H, 1) (H, 2) (H, 3) (H,4 ) (H, 5) (H, 6)

(T, 1) (T, 2) (T, 3) (T, 4) (T, 5) (T, 6)

TREE DIAGRAM

The “tree” is constructed from the left to the right. A “tree diagram” is a useful device for enumerating all the possible outcomes of two or more sequential events. The possible outcomes are represented by the individual paths or branches of the tree. It is relevant to note that, in general

A × B ≠ B × A. Having reviewed the basics of set theory, let us now review the COUNTING RULES that facilitate the computation of probabilities in a number of problems.

RULE OF MULTIPLICATION

If a compound experiment consists of two experiments which that the first experiment has exactly m distinct outcomes and, if corresponding to each outcome of the first experiment there can be n distinct outcomes of the second experiment, then the compound experiment has exactly mn outcomes.

EXAMPLE

The compound experiment of tossing a coin and throwing a die together consists of two experiments. The cointossing experiment consists of two distinct outcomes (H, T), and the diethrowing experiment consists of six distinct outcomes (1,2,3,4, 5, 6).

The total number of possible distinct outcomes of the compound experiment is therefore 2 × 6 = 12 as each of the two outcomes of the cointossing experiment can occur with each of the six outcomes of diethrowing experiment. As stated earlier, if A = {H, T} and B = {1, 2, 3, 4, 5, 6}, then the Cartesian product set is the collection of the following twelve (2 × 6) ordered pairs: A×B = { (H, 1); (H, 2);(H, 3); (H, 4);

(H, 6); (H, 6);(T, 1); (T, 2); (T, 3); (T, 4); (T, 5); (T, 6) }

Tree Diagram

A BA B

statistics and probability  Set Theory

statistics and probability  Set Theory

1 1 1 1 1 1

1 2 3 4 5 6 (H, 1) (H, 2) (H, 3) (H,4 ) (H, 5) (H, 6)

(T, 1) (T, 2) (T, 3) (T, 4) (T, 5) (T, 6)

The rule of multiplication can be readily extended to compound experiments consisting of any number of experiments performed in a given sequence.

This rule can also be called the Multiple Choice Rule, as illustrated by the following example:

EXAMPLE

Suppose that a restaurant offers three types of soups, four types of sandwiches, and two types of desserts. Then, a customer can order any one out of 3 × 4 × 2 = 24 different meals.

EXAMPLE

Suppose that we have a combination lock on which there are eight rings. In how many ways can the lock be adjusted?

Solution:

The logical way to look at this problem is to see that there are eight rings on the lock, each of which can have any of the 10 figures 0 to 9:

ABC DEFGH

ring A can have any of the digits 0 to 9 and ring B can have any of the digits 0 to 9 and ring C can have

any of the digits 0 to 9 and . . .

ring H can have any of the digits 0 to 9 Hence the total No. of ways in which these 8 rings can be filled is 8 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 = 10

i.e. 100,000,000 –– one hundred million.

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