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Subsets, Proper Subset, Improper Subset, Intervals - (Concept)
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 $\}$
| Exam | Chapter |
| JEE MAIN | Sets, Relations and Functions |
Let A and B be two sets containing four and two elements respectively. Then the number of subsets of the set A x B, each having at least three elements is :
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If A={1,2,3,5,7} and B={3,5}, then
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If A is a proper subset of B, we write.
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Let A and B be two sets containing 2 elements and 4 elements respectively. The number of subsets of A $\times$ B having 3 or more elements is:
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Let $\mathrm{A}, \mathrm{B}$ and C be sets such that $\phi \neq A \cap B \subseteq C$. Then which of the following statements is not true?
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If a set has 256 subsets. How many elements does it have?
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The number of non-empty subsets of is:
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Let The number of non-empty subsets A of S such that the product of elements in A is even is :
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Set A has m elements and set B has n elements. If the total number of the subset of A is 112 more than the total number of subsets of B, then (m - n) equals.
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Find the number of subsets of the set A={5,7}.
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Select the proper subset of set A={3,5,7}.
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If a set A has 8 elements, then the number of proper subsets of the set A is
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Which of the following sets does not have a proper subset?
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Two finite sets have m and n elements. The total number of subsets of the first set are 56 more than the second set, find m and n.
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If $A=\{x \in R:|x-2|>1\}, B=\left\{x \in R: \sqrt{x^2-3}>1\right\}$, $C=\{x \in R:|x-4| \geq 2\}$ and $\mathbf{Z}$ is the set of all integers, then the number of subsets of the set $(A \cap B \cap C)^c \cap Z$ is____________.
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Two sets A and B are as under : A={(a, b) R×R :| a−5| < 1 and |b−5| < 1}; B={(a, b)
R×R : 4(a−6)2+9(b−5)2 ≤ 36}. Then :
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If a set has 32 subsets. How many elements does it have?
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Two finite sets have m and n elements. The total number of subsets in the first set is 56 more than the second set. Find m and n.
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The number of non-empty subsets of A={a,b,c,d,e} is:
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Which of the following intervals is the largest?
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Which of the following is correct representation of the interval [5,7] on number line?
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Which of the following Venn diagram best explains the sets of Dogs, Cats and Animals?
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If P = {1, 3, 5, 7, 9, 11, 13}, then which of the following is subset of P?
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$(2,4) \cup(3,10)$ equals
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Union of the intervals (-2,0), (3,5) and [-3,12) equals
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Intersection of intervals $(2,5)$ and $(\pi, 4]$ is
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$(-3,4) \cap[0,5] \cap(5,7]$ equals
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If , then number of subsets of
is.
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Let the number of elements in sets A and B be five and two respectively. Then the number of subsets of A × B each having at least 3 and at most 6 elements is :
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Let A and B be two finite sets with m and n elements respectively. The total number of subsets of the set A is 56 more than the total number of subsets of B. Then the distance of the point P(m, n) from the point Q(–2,–3) is:
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If sets A and B are defined as
$A= \left \{ (x,y) : y = e ^{2x}, x \epsilon R \right \} \: \: and \: \: B = \left \{ (x,y ): y = x -1 , x \epsilon R \right \}$,then:
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All the subsets of $\left \{ a, b,c \right \}$ are
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For a set having 10 elements, no.of subsets are
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How do we write: That A is a proper subset of B?
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How should we write: "All real numbers between 1 and 2 such that 2 is included and 1 is excluded"?
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Identify the subset of $A=\{2,3,5,7,11,13\}$ from the following:
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If a set A has n elements, then the total number of subsets of A is:
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The number of proper subsets of set {1, 2, 3} is:
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Two finite set have m and n two element . The total no. of subset of first set is 48 more than the total no. of subset of second set .Then (m,n) are
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If A = $\{x: x=4 n+1,2 \leq n \leq 5\}$, then number of subset of A is:
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Two finite sets have m and n elements. The number of subsets of the first set is 112 more than that of the second set. The values of m and n are, respectively:
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Let $A=\{1,2,3,4,5\} \& A_i \subseteq A$ for $i=1,2,3,4$. Find the number of ways in which the subsets $A_1, A_2, A_3 \& A_4$ can be chosen such that $A_i \cap A_j=\phi$ for all $i \neq j$.
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If $A=\left \{ 1,2,\left \{ 3,4 \right \} \right \},$ then which of the following is true.
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$\text { Find the no. of subsets of a set } A=\left\{x: x \in \text { Natural Number } \forall x^3<100\right\}$
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Which of the following statements is true ?
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The number of subsets of a set containing $\mathrm{n}$ distinct object is
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Let $E$ denote the sets of letters of the English alphabet, $V=\{a, e, i, o, u\}$ and $C$ be the complement of $V$ in $E$.Then,the number of four-letter words (where repetitions of letters are allowed) having at least one letter from $V$ and at least one letter from $C$ is
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Assertion: Any selection from a collection of unique items yields a subset of the entire set
Reason: Selecting any number of different items from a set entails choosing just some members from the original set, which results in the creation of a subset, which is defined as a set having only components from a larger set.
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If $A = {1,2,3,4,5,6} $ then the number of subsets of A which contain at least two elements is:
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A set A is said to be a subset of a set B if every element of A is also an element of B.
