Dxc

Sets

By- Sahil Suman

Class- XI-D

Introduction• The theory of sets was

developed by German mathematician George Cantor.

• A set is a collection of objects.• Objects in the collection are

called elements of the set.• They are named by capital

English alphabet.

Representation Of Sets • Roster form and Set Builder form

• Roster Form- when the elements are written inside the set It is defined as a set by actually listing its elements, for example, the elements in the set A of letters of the English alphabet can be listed as A={a,b,c,……….,z} separated by comas.

• Set Builder Form- when we write a set in a straight form using underlying relations that binds them.

• Example- {x | x < 6 and x is a counting number} in the set of all counting numbers less than 6. Note this is the same set as {1,2,3,4,5}.

Types Of Sets• Empty Sets• Finite Sets• Infinite Sets• Equal Sets• Subsets• Power Sets• Universal Sets

Empty Sets• A set that contains no members

is called the empty set or null set .

• For example, the set of the months of a year that have fewer than 15 days has no member .Therefore ,it is the empty set. The empty set is written as { } or .

Finite Sets• A set is finite if it consists of a

definite number of different elements ,i.e., if in counting the different members of the set, the counting process can come to an end.

• For example, if W be the set of people living in a town, then W is finite.

Infinite Sets•  An infinite set is a set that is not

a finite set. Infinite sets may be countable or uncountable. Some examples are:

• The set of all integers, {..., -1, 0, 1, 2, ...}, is a count ably infinite set;

Equal Sets• Equal sets are sets which have the

same members. Or Two sets a and b are said to be equal if they have the same no of elements.

• For example, if P

={1,2,3},Q={2,1,3},R={3,2,1} then P=Q=R.

Subsets• Sets which are the part of another set

are called subsets of the original set.• For example, if

A={1,2,3,4} and B ={1,2}then B is a subset of Ait is represented by .

Power Sets• If ‘A’ is any set then one set of all are subset of set ‘A’ that it

is called a power set.• Example- If S is the set {x, y, z}, then the subsets of S are:• {} (also denoted , the empty set)• {x}• {y}• {z} • {x, y}• {x, z}• {y, z}• {x, y, z}• and hence the power set of S is {{}, {x}, {y}, {z}, {x, y},

{x, z}, {y, z}, {x, y, z}}.

Universal Sets • A universal set is a set which contains all

objects, including itself. Or• In a group of sets if all the sets are the subset

of a particular bigger set then that bigger set then that bigger set is called the universal set.

• Example- A={12345678} B={1357} C={2468} D={2367} Here A is universal set and is denoted by

Operation Of Sets• Union of sets• Intersection of sets• Compliments of sets

Union• The union of two sets would be

wrote as A U B, which is the set of elements that are members of A or B, or both too.

• Using set-builder notation, A U B = {x : x is a member of A

or X is a member of B}

Intersection• Intersection are written as A ∩ B,

is the set of elements that are in A and B.

• Using set-builder notation, it would look like:

A ∩ B = {x : x is a member of A and x is a member of B}.

Complements• If A is any set which is the subset of a

given universal set then its complement is the set which contains all the elements that are in

but not in A.• Notation A’ ={1,2,3,4,5} A={1,2,3} A’={2,4}

Some Other Sets• Disjoint – If A ∩ B = 0, then A and B

are disjoint.• Difference: B – A; all the elements in

B but not in A• Equivalent sets – two sets are

equivalent if n(A) = n(B).

Venn Diagrams• Venn diagrams are

named after a English logician, John Venn.

• It is a method of visualizing sets using various shapes.

• These diagrams consist of rectangles and circles.

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