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Universal Set

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Universal Set - Lesson Summary

  Subset
 
A set P is said to be a subset of set Q if every element of P is also an element of Q.

This is symbolically represented as P ⊂ Q, if x ∈ P ⇒ x ∈ Q

P is a subset of Q if x is an element of P, it implies that x is also an element of Q.

P is not a subset of Q.
P ⊄ Q

1) If every element of set P is contained in Q and vice-versa, then sets P and Q are equal.

      This can be defined symbolically as P ⊂ Q , Q ⊂ P ⇔ P = Q.

2) Every set is a subset of itself.
 
     This is symbolically written as ⊂ A .

3) ⊂ B and A ≠ B
     A is a proper subset of B.
     B is a superset of A.

4) If any set P contains only one element, then P is called a singleton set.
Example: {5},{x}
 
5) An empty set is a subset of every set.
 
Universal Set

Consider S = {1, 2, 3}

{1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3} are the subsets of the set S.
 
The set containing these sets as elements is called the power set of set S and is denoted by P(S).
 
P(S) = {{1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3}}
 
m[P(S)] = 8
 
If a set S consists of m number of elements, the number of elements of the power set is equal to 2 m.
 
Consider B = {3,6}
 
P(B) = {{3},{6},{3,6}}

n(B) = 2

n[P(B)] = 22 = 4
 
The set of natural numbers is a part of another set, that is, the set of real numbers.

In this context, the set of real numbers is the universal set.

Similarly, for the set of integers, the universal set is the set of real numbers.
 
The set of natural numbers is a subset of whole numbers.

The set of rational numbers is a subset of the set of real numbers.

Consider the real numbers in the interval between the numbers two and three.

The first interval includes the numbers two and three.

This interval consists of all the numbers between the numbers two and three.

In set-builder notation, the interval is represented as [2,3] = {x:2 ≤ x ≤ 3}.

The second interval is all the numbers within the interval two and three, excluding the number two.
 
This is represented in set-builder notation as (2,3] = {x:2 < x ≤ 3}.
 
The third interval is all the numbers within the interval two and three excluding three.

This interval is represented in the set-builder form as [2,3) = {x:2 ≤ x < 3}.

The fourth interval is all the numbers within the interval two and three, excluding two and three.

This interval is represented in the set-builder form as (2,3) = {x:2 < x < 3}.
 
Some sets of real numbers.

The interval zero to infinity with zero included is the set of non-negative real numbers.
 
Set of non-negative real numbers = [0, ∞)

Set of negative real numbers = (-∞, 0)

Set of real numbers = (-∞, ∞)
 
On generalising, we can conclude that
 
[a, b] = {x: a ≤ x ≤ b, a, b ∈ R}

(a, b] = {x: a < x ≤ b, a, b ∈ R}

(a, b) = {x: a < x < b, a, b ∈ R}

[a, b) = {x: a ≤ x < b, a, b ∈ R}

(b-a) is the length of the intervals (a, b), (a, b], [a, b) and [a, b].

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