Mathematics Subsets , Intervals as subsets of R , Power Set and Universal Set
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### Topics Covered

star Subsets , Proper subset and Superset
star Intervals as subsets of R

### Subsets , Proper subset and Superset

Consider the sets : X = set of all students in your school, Y = set of all students in your class.

• We note that every element of Y is also an element of X; we say that Y is a subset of X.

• The fact that Y is subset of X is expressed in symbols as Y ⊂ X. The symbol ⊂ stands for "‘is a subset of’" or "‘is contained in’."

\color{purple}ul(✓✓) \color{purple} " DEFINITION ALERT"
A set A is said to be a subset of a set B if every element of A is also an element of B.

• In other words, A ⊂ B if whenever a ∈ A, then a ∈ B. It is often convenient to use the symbol "“⇒”" which means implies. Using this symbol, we can write the definition of subset as follows:

 \ \ \ \ \ \ \ \ \ \ A subset B if a ∈ A ⇒ a ∈ B

• We read the above statement as “A is a subset of B. if a is an element of A implies that a is also an element of B”.

\color{blue} (✍️ "If A is not a subset of B, we write A ⊄ B").

\color{green} ✍️ \color{green} \mathbf(KEY \ CONCEPT)
• It follows from the above definition that every set A is a subset of itself, i.e., A subset A.
• Since the empty set φ has no elements, we agree to say that φ is a subset of every set.

"We now consider some examples :"
(i) The set Q of rational numbers is a subset of the set R of real numbes, and we write Q ⊂ R.

(ii) If A is the set of all divisors of 56 and B the set of all prime divisors of 56, then B is a subset of A and we write B ⊂ A.

(iii) Let A = {1, 3, 5} and B = {x : x "is an odd natural number less than" 6}. Then A ⊂ B and B ⊂ A and hence A = B.

(iv) Let A = { a, e, i, o, u} and B = { a, b, c, d}. Then A is not a subset of B, also B is not a subset of A.

\color{green} (✍️ "Proper subset and Superset")
Let A and B be two sets. If A ⊂ B and A ≠ B , then A is called a proper subset of B and B is called superset of A.

For example, A = {1, 2, 3} is a proper subset of B = {1, 2, 3, 4}.

\color{green}( ✍️ "Singleton set: ")
If a set A has only one element, we call it a singleton set. Thus ,{ a } is a singleton set.

Q 3087367287

Consider the sets
phi, A = { 1, 3 }, B = {1, 5, 9}, C = {1, 3, 5, 7, 9}.
Insert the symbol ⊂ or ⊄ between each of the following pair of sets:

(i) phi . . . B (ii) A . . . B (iii) A . . . C (iv) B . . . C

Solution:

(i) phi ⊂ B as phi is a subset of every set.
(ii) A ⊄ B as 3 ∈ A and 3 ∉ B
(iii) A ⊂ C as 1, 3 ∈ A also belongs to C
(iv) B ⊂ C as each element of B is also an element of C.
Q 3087467387

A = { a, e, i, o, u} and B = { a, b, c, d}.

(1) . Is A a subset of B ?

(2) . Is B a subset of A?

Solution:

(1) . No
(2) . No
Q 3017567480

Let A, B and C be three sets. If A ∈ B and B ⊂ C, is it true that A ⊂ C?. If not, give an example.

Solution:

No. Let A = {1}, B = {{1}, 2} and C = {{1}, 2, 3}. Here A ∈ B as A = {1} and B ⊂ C. But A ⊄ C as 1 ∈ A and 1 ∉ C.
Note that an element of a set can never be a subset of itself.

### Intervals as subsets of R

\color{green} ✍️ \color{green} \mathbf(KEY \ CONCEPT)
Let a, b ∈ R

• a < b  then the set of real numbers { y : a < y < b} is called an "open interval" and is denoted by (a, b).

• All the points between a and b belong to the open interval (a, b) but a, b themselves do not belong to this interval.

• The interval which contains the end points also is called "closed interval" and is denoted by [ a, b ]. Thus
 \ \ \ \ \ \ \ \ \ [ a, b ] = {x : a ≤ x ≤ b}

• We can also have intervals closed at one end and open at the other, i.e.,
[ a, b ) = {x : a ≤ x < b} is an open interval from a to b, including a but excluding b.
( a, b ] = { x : a < x ≤ b } is an open interval from a to b including b but excluding a.

These intervals are also known as semi open or semi closed or half open or half closed.
[a,b) is also written as  [a,b[

\color{green} ✍️ \color{green} \mathbf(KEY \ CONCEPT)
(a,b] is also written as  ]a,b]

• These notations provide an alternative way of designating the subsets of set of real numbers.
For example , if A = (–3, 5) and B = [–7, 9], then A ⊂ B.

On real number line, various types of intervals described above as subsets of R, are shown in the Fig.

• Here, we note that an interval contains infinitely many points.

\color{blue} (✍️ "The number (b – a) is called the length of any of the intervals") (a, b), [a, b], [a, b) or (a, b].
Q 1904145958

Write the following as intervals :

(i) {x : x in R , - 4 < x le 6}

(ii) {x : x in R, - 12 < x < -10}

(iii) {x : x in R, 0 le x < 7}

(iv) {x : x in R, 3 le x le 4}
Class 11 Exercise 1.3 Q.No. 6
Solution:

(i) (-4,6]

(ii) (-12,-10 )

(iii) [0 , 7)

(iv) [3,4]