Sets - Sets, Class 11, Mathematics PDF Download

SET

A set is a well-defined collection of distinct objects. Well-defined collection means that there exists a rule with the help of which it is possible to tell whether a given object belongs or does not belong to given collection. Generally sets are denoted by capital letters A, B, C, X, Y, Z etc.

REPRESENTATION OF A SET

Usually, sets are represented in the following ways:

ROASTERFORMOR TABULARFORM

In this form, we list all the member of the set within braces ≤  curly brackets) and separate these by commas. For example, the set of all even numbers less than 10 and greater than 0 in the roster form is written as: A = {2,4, 6,8}

SET BUILDERFORMOR RULEFORM

In this form, we write a variable ≤ say x) representing any member of the set followed by a property satisfied by each member of the set.A = {x| x £ 5, x ∈ N} the symbol ‘|’ stands for the words” such that”.

NULL/ VOID/ EMPTY SET

A set which has no element is called the null set or empty set andis denoted by ϕ ≤ phi). The number of elements of a set A is denoted as n ≤ A) and n ≤ ϕ) = 0 as it contains no element. For example the set of all real numbers whose square is –1.

SINGLETON SET

A set containing only one element is called Singleton Set.

FINITEANDINFINITE SET

A set, which has finite numbers of elements, is called a finite set. Otherwise it is called an in finite set. For example, the set of all days in a week is a finite set whereas; the set of all integers is an infinite set.

UNION OF SETS

Unionof 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: 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}

DIFFERENCE OF 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}. 

Symmetric Difference of Two Sets:

For two sets A and B, symmetric difference of A and B is given by ≤ A – B) ∪ ≤ B – A) and is denoted by A Δ B.