Subsets, Proper Subset, Improper Subset, Intervals - Practice Questions & MCQ

Set

A set is a well-defined collection of distinct objects and it is usually denoted by capital letters A, B, C, S, U, V......

Subset

A set A is said to be a subset of a set B if all elements of A are present in B.

It is represented by ⊂ .

A = {1, 2, 3} ,       B = {1, 2, 3, 4, 5} ,   C = {3, 2, 1}

A is a subset of B  or A ⊂ B

C is a subset of A or C ⊂ A

Also, if all elements of set A present in set B, then A is a subset of B and B is called a superset of A

Properties

1. If sets A and B are subsets of each other then they are equal sets.

    eg, If A = {1, 2, 3} ,  B = {3, 1, 2},

    Here, A ⊂ B and B ⊂ A  ⇒ A = B

2. Every set is a subset of itself, A ⊂ A.

3. φ is a subset of every set.

Proper and Improper Subsets

If A is a subset of B but , then we say A is a proper subset of B.

And if , then we say that A is an improper subset of B. 

Example:  If A={2,4} and B= {1,2,3,4,5}, then A is a proper subset of B

And C = {1,2,3,4,5} is improper subset of B in this case

Note:

1. Every set has one improper subset, all other subsets are proper subsets.

2. φ has only one subset, which is φ itself. So, φ does not have any proper subset.

3. Important sets related to numbers

N : the set of all natural numbers 

Z : the set of all integers 

Q : the set of all rational numbers

Q': the set of all irrational numbers

R : the set of real numbers 

Z⁺ : the set of positive integers 

Q⁺ : the set of positive rational numbers 

R⁺ : the set of positive real numbers.

N ⊂ Z ⊂ Q ⊂ R

Number of subsets of a set
If a set $A$ has $n$ elements, then the total number of subsets of $A$ is $2^n$.
Also as each subset has one improper subset, so number of proper subsets is $\left(2^n-1\right)$.

Example: $A=\{1,2,3\}$
All the objects that form a set are called its elements or members. These are usually denoted by small letters, i.e. $x, y, z \ldots$.

If x is an element of a set A , we write $\mathrm{x} \in \mathrm{A}$ and read as ' x belongs to A '.
If $x$ is not an element of a set $A$, we write $x \notin A$ and read it as ' $x$ does not belong to $A$ '.
Example: $A=\{1,2,3\}$, then $2 \in A$ ( 2 belongs to set $A$ ) and $4 \notin A$ ( 4 does not belong to set $A$ )
There are two methods of representing a set - Roster (or Tabular) form \& Set-builder Form.

Roster or Tabular form

In roster form, all the elements of a set are listed, the elements are separated by commas and are enclosed within braces \{ \}.

Example: $\{\mathrm{a}, \mathrm{e}, \mathrm{i}, \mathrm{o}, \mathrm{u}\}$ represents the set of all the vowels in the English alphabet in the roster from.
In roster form, the order in which the elements are listed is immaterial, i.e. the set of all natural numbers which divide 14 is $\{1,2,7,14\}$ can also be represented as $\{1,14,7,2\}$.

An element is not generally repeated in the roster form of a set, i.e., all the elements are taken as distinct. For example, the set of letters forming the word 'SCHOOL' is $\{\mathrm{S}, \mathrm{C}, \mathrm{H}, \mathrm{O}, \mathrm{L}\}$ or $\{\mathrm{H}, \mathrm{O}, \mathrm{L}, \mathrm{C}$, S\}. Here, the order of listing elements has no relevance.

Set-builder Form

In set-builder form, all the elements of a set possess a single common property that is not possessed by any element outside the set. If $Z$ contains all values of $x$ for which the condition $q(x)$ is true, then we write

$
Z=\{x: q(x)\} \text { or } Z=\{x \mid q(x)\}
$
Where, ': ' or ' $\mid$ ' is read as 'such that'
eg. The set $A=\{0,1,8,27,64, \ldots$.$\} can be written in Set Builder form as$
$A=\left\{x^3: x\right.$ is a nonnegative integer $\}$