Sets
- A set is a well-defined collection of elements. These elements may include numbers, symbols, variables, etc.
- We generally represent them using capital letter(A, B, C, D, Z, N etc).
- Elements in the sets are represented by small letters(a, b, c, d, e)...
Representation of Sets
To define sets, we also need to learn different representations of sets.
There are three main forms of set representations.
1. Roster or Tabular Form
In this form or representation of a set, we simply list all the elements. Commas separate them. All the elements are within curly brackets.
Example: A={2,4,6,8,10}
Important: The order of elements does not matter, and duplicates are not written.
2. Set-builder Form
In this form, a set is described by a property that its elements satisfy.
Example: A={x:x is an even natural number less than 12 }
This is read as ' A is the set of all x such that x is an even natural number less than 12.'
Types of Sets
1. Empty Set or Null Set: A set with no elements. Denoted by } or ∅ .
- Example: A={x:x is a natural number less than 1 }
2. Finite Set: A set with a finite number of elements.
- Example: B = {1, 2, 3, 4, 5}
3. Infinite Set: A set with an uncountable or infinite number of elements.
- Example: C={x:x is a natural number }
4. Equal Sets: Two sets A and B are equal if they have exactly the same elements.
- Example: A={1,2,3},B={3,1,2}⇒A=B
5. Equivalent Sets: Two sets are equivalent if they have the same number of elements. It does not depend on what the elements are.
- Example: A={1,2,3},B={a,b,c}⇒A and B are equivalent
6. Singleton Set: A set with only one element.
- Example: A = {0}
7. Subset: A is a subset of B if every element of A is also in B . Denoted by A⊆B .
8. Proper Subset : If A⊆B and A≠B , then A is a proper subset of B . Denoted by A⊂B .
9. Power Set : The set of all subsets of a given set A. Denoted by P(A) .
10. Universal Set : A set that contains all elements under consideration. Denoted by U.
11. Disjoint Sets: Two sets with no common elements.
- Example: A={1,2},B={3,4}⇒A∩B=∅
Cardinality of a Set
So, what is a cardinality of a set?
The number of elements in a set is called the cardinality of the set. It's also known as the size of the set. It is denoted by |A| for a set A.
Example: If A={1,2,3,4} , then |A|=4 .
When we talk about set notation, it is what one set is part of. In this case, they are within curly brackets { }.
We keep the elements within specific symbols. This is to define the set's elements. Or, it could be to how two sets are linked.
∈ : Element of
∉: Not an element of
⊆ : Subset of
C : Proper subset of
⊄: : Not a subset of
∅ or } : Empty set
|A| : Cardinality of set A
N : Set of natural numbers ={1,2,3,…}
W : Set of whole numbers ={0,1,2,3,…}
Z : Set of integers ={…,-2,-1,0,1,2,…}
Q : Set of rational numbers
R : Set of real numbers
C : Set of complex numbers