Set Theory

Set:

A set is a well-defined collection of objects or elements. Each element in a set is unique.

Usually but not necessarily a set is denoted by a capital letter e.g. A, B, … U, V etc. and the elements are enclosed between brackets { }, denoted by small letters a , b , ….x , y etc.

For example:

A = Set of all small English alphabets

= {a , b , c ,… , x , y , z}

B = Set of all positive integers less than or equal to 10

= {1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10}

R = Set of real numbers = {x: − ∞ < x < ∞}

The elements of a set can be discrete (e.g. set of all English alphabets) or continuous (e.g. set of real numbers).

The set may contain finite or infinite number of elements. A set may contain no elements and such a set is called Void set or Null set or empty set and is denoted by φ (phi).

The number of elements of a set A is denoted as n(A) and hence n( φ) = 0 as it contains no element.

Union of Sets:

Union of two or more sets is the set of all elements that belong to any of these sets. The symbol used for union of sets is ‘ ∪ ‘

i.e. A∪B = Union of set A and set B

= {x: x ∈ A or x∈B (or both)}

Example:

If A = {1, 2, 3, 4} and B = {2, 4, 5, 6} and C = {1, 2, 6, 8}, then A∪B∪C = {1, 2, 3, 4, 5, 6, 8}

Intersection of Sets:

It is the set of all the elements, which are common to all the sets.
The symbol used for
intersection of sets is ‘ ∩ ‘.

i.e. A∩B = {x: x ∈ A and x∈ B}

Example:
If A = {1, 2, 3, 4} and B = {2, 4, 5, 6} and C = {1, 2, 6, 8}, then A ∩ B ∩ C = { 2 }
Remember that n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

Difference of two sets:

The difference of set A to B denoted, as A – B is the set of those elements that are in the set A but not in the set B

i.e. A – B = {x: x ∈ A and x ∉ B}.
Similarly B − A = {x: x ∈ B and x ∉ A}.

In general A − B ≠ B − A.
Example:
If A = {a, b, c, d} and B = {b , c , e , f}

then A − B = {a, d} and B − A = {e , f}

Subset of a Set:

A set A is said to be a subset of the set B if each element of the set A is also the element of the set B. The symbol used is ‘ ⊆ ‘

i.e. A ⊆ B <=> (x ∈ A

=> x ∈ B)

Each set is a subset of its own set. Also a void set is a subset of any set.

If there is at least one element in B which does not belong to the set A, then A is a proper subset of set B and is denoted as A ⊂ B.

If set B has n elements then total number of subsets of set B is 2n

Example:

If A = {a, b, c, d} and B = {b, c, d} then B ⊂ A or equivalently A ⊃ B (i.e A is a super set of B).

Equality of Two Sets:

Sets A and B are said to be equal if A ⊆ B and B ⊆ A and we write A = B.

Universal Set:

As the name implies, it is a set with collection of all the elements and is usually denoted by U.

e.g. set of real numbers R is a universal set whereas a set A ={x: x ≤ 3} is not a universal set as it does not contain the set of real numbers x >3.

Once the universal set is known, one can define the Complementary set of a set as the set of all the elements of the universal set which do not belong to that set.
e.g. If A = {x: x ≤ 3} then A˜ = complimentary set of A = {x: x > 3}.

Hence we can say that A ∪ A˜ = U i.e. Union of a set and its complimentary is always the Universal set and A ∩ A˜ = φ

i.e intersection of the set and its complimentary is always a void set.

Cartesian product of Sets:

The Cartesian product (also known as the cross product) of two sets A and B , denoted by A x B (in the same order) is the set of all ordered pairs (x , y) such that x ∈ A and y ∈ B.

What we mean by ordered pair is that the pair (a, b) is not the same pair as (b, a) unless a = b. It implies that

A x B ≠ B x A in general. Also if A contains m elements and B contains n elements then A x B contains m x n elements.

Similarly we can define A x A = {(x, y); x ∈ A and y ∈ A}. We can also define cartesian product of more than two sets.

e.g. A1 x A2 x A3 x . . . . x An = {(a1, a2, . . . , an) : a1 ∈ A1 , a2 ∈ A2, . . . , an ∈ An}

Example: 1.

Sixty five percent of children in a sport club play football , 70 percent play volley-ball and 75 per-cent play basket ball. What is the smallest percentage of children playing all the three games?

Solution: Out of 100 children,

Number of children who do not play football = 100 − 65 = 35
Number of children who do not play volley-ball = 100 − 70 = 30
Number of children who do not play basket ball = 100 − 75 = 25.
So, the maximum number of children who do not play atleast one game
= 35 + 30 + 25 = 90.

Thus the minimum number of children who play all the three games = 100 − 90 = 10.

Hence the smallest percentage of children playing all the three games = 10%.

Note: The greatest percentage of children playing all the three games = min(65%, 70%, 75%} = 65%.

NUMBER THEORY:

Natural Numbers:

The numbers 1 , 2 , 3 , 4 … are called natural numbers, their set is denoted by N. Thus N = {1 , 2 , 3 , 4 , 5 …}

Integers:

The numbers … −3 , −2 , −1 , 0 , 1 , 2 , 3 … are called integers and the set is denoted by I or Z.
Thus I (or Z) = {… −3 , −2 , −1, 0,1, 2, 3 …}.

Remarks:

⋄ Integers 1 , 2 , 3 , … are called positive integers or natural number and denoted by I+ or N.

⋄ Integers …, −3, −2, −1 are called negative integers and denoted by I

⋄ Integers 0 , 1 , 2 , 3 , … are called whole numbers or non-negative integers.

⋄ Integers …, −3, −2, −1, 0 are called non-positive integers

Rational Numbers:

The numbers which can be expressed in the form p/q where p and q are integers, H.C.F. of p and q is 1 and q ≠ 0, are called rational numbers and their set is denoted by Q.

Thus Q = { p/q : p , q ∈I and q ≠0 and HCF of p , q is 1 }.

It may be noted that every integer is a rational number since it can be written as p/1.

It may also be noted that all recurring decimals are rational numbers. e.g., p = = 0.33333…

Then, 10p − p = 3

=> p = 1/3 , which is a rational number.

Irrational Numbers:

There are numbers, which can not be expressed in p/q form. These numbers are called irrational numbers and their set is denoted by Qc (i.e. complementary set of Q)

e.g. √2 , 1 +, √3 , π , e , √5 etc.

Irrational numbers can not be expressed as terminating decimals or recurring decimals.

Real Numbers:

The complete set of rational and irrational numbers is the set of real numbers and is denoted by R. Thus R = Q ∪ QC

It may be noted that N⊂ I⊂ Q⊂ R. The real numbers can also be expressed in terms of the positions of a point on the straight line.

The straight line is defined as the real number line wherein the position of a point relative to the origin (i.e. 0) represents a unique real number and vice versa.

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