Mathematics Class 11
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Subsets, Intervals, and Power Sets

🎓 Class 11 Mathematics CBSE Theory Ch 1 — Sets ⏱ ~25 min
🌐 Language: [gtranslate]

This MCQ module is based on: Subsets, Intervals, and Power Sets

This mathematics assessment will be based on: Subsets, Intervals, and Power Sets
Targeting Class 11 level in Sets, with Advanced difficulty.

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1.4 Finite and Infinite Sets

Consider the sets: \(A = \{1, 2, 3, 4, 5\}\), \(B = \{a, b, c, d, e, g\}\), and \(C = \{\text{men living presently in different parts of the world}\}\).

We observe that A contains 5 elements and B contains 6 elements. How many elements does C contain? As it is, we do not know the exact number of elements in C, but it is some (very large) natural number. By number of elements of a set S, we mean the number of distinct elements and we denote it by \(n(S)\). If \(n(S)\) is a natural number, then S is called a non-empty finite set?.

Now consider the set of natural numbers. The number of elements of this set is not finite since there are infinitely many natural numbers. We say that the set of natural numbers is an infinite set?. The sets A, B, and C given above are finite sets and \(n(A) = 5\), \(n(B) = 6\), and \(n(C) =\) some finite number.

Definition
A set which is empty or consists of a definite number of elements is called finite; otherwise, the set is called infinite.

Consider some examples:

  1. Let W be the set of the days of the week. Then W is finite.
  2. Let S be the set of solutions of the equation \(x^2 - 16 = 0\). Then \(S = \{4, -4\}\), which is finite.
  3. Let G be the set of points on a line. Then G is infinite.
Note
All infinite sets cannot be described in the roster form. For example, the set of real numbers cannot be described in this form, because the elements of this set do not follow any particular pattern.

Example 6

State which of the following sets are finite or infinite:

(i) \(\{x : x \in \mathbb{N} \text{ and } (x-1)(x-2) = 0\}\)   (ii) \(\{x : x \in \mathbb{N} \text{ and } x^2 = 4\}\)   (iii) \(\{x : x \in \mathbb{N} \text{ and } 2x - 1 = 0\}\)   (iv) \(\{x : x \in \mathbb{N} \text{ and } x \text{ is prime}\}\)   (v) \(\{x : x \in \mathbb{N} \text{ and } x \text{ is odd}\}\)

Solution
(i) Given set = \(\{1, 2\}\). Hence, it is finite.
(ii) Given set = \(\{2\}\) (since \(x \in \mathbb{N}\)). Hence, it is finite.
(iii) Given set = \(\emptyset\) (since \(x = \frac{1}{2} \notin \mathbb{N}\)). Hence, it is finite.
(iv) The given set is the set of all prime numbers and since the set of prime numbers is infinite, the given set is infinite.
(v) Since there are infinite number of odd numbers, the given set is infinite.

1.5 Equal Sets

Given two sets A and B, if every element of A is also an element of B and if every element of B is also an element of A, then the sets A and B are said to be equal?. Clearly, the two sets have exactly the same elements.

Definition
Two sets A and B are said to be equal if they have exactly the same elements and we write \(A = B\). Otherwise, the sets are said to be unequal and we write \(A \neq B\).

Examples

  1. Let \(A = \{1, 2, 3, 4\}\) and \(B = \{3, 1, 4, 2\}\). Then \(A = B\).
  2. Let A be the set of prime numbers less than 6 and P the set of prime factors of 30. Then \(A = \{2, 3, 5\}\) and \(P = \{2, 3, 5\}\). Thus \(A = P\).
Note
A set does not change if one or more elements of the set are repeated. For example, the sets \(A = \{1, 2, 3\}\) and \(B = \{2, 2, 1, 3, 3\}\) are equal, since each element of A is in B and vice-versa. This is why we generally do not repeat any element in describing a set.

Example 7

Find the pairs of equal sets, if any, and give reasons:

\(A = \{0\}\), \(B = \{x : x \gt 15 \text{ and } x \lt 5\}\), \(C = \{x : x - 5 = 0\}\), \(D = \{x : x^2 = 25\}\), \(E = \{x : x \text{ is an integral positive root of the equation } x^2 - 2x - 15 = 0\}\)

Solution
Since \(0 \in A\) and 0 does not belong to any of the sets B, C, D and E, it follows that \(A \neq B\), \(A \neq C\), \(A \neq D\), \(A \neq E\).

Since \(B = \emptyset\) but none of the other sets are empty: \(B \neq C\), \(B \neq D\), \(B \neq E\).

Since \(E = \{5\}\), \(C = E\). Further, \(D = \{-5, 5\}\) and \(E = \{5\}\), we find that \(D \neq E\).

Thus, the only pair of equal sets is C and E.

Example 8

Which of the following pairs of sets are equal? Justify your answer.

(i) X, the set of letters in "ALLOY" and B, the set of letters in "LOYAL".
(ii) \(A = \{n : n \in \mathbb{Z} \text{ and } n^2 \leq 4\}\) and \(B = \{x : x \in \mathbb{R} \text{ and } x^2 - 3x + 2 = 0\}\).

Solution
(i) We have, \(X = \{A, L, I, O, Y\}\), \(B = \{L, O, Y, A, L\}\). Since repetition of elements in a set does not change a set, \(B = \{L, O, Y, A\}\). Since \(I \in X\) but \(I \notin B\), \(X \neq B\). Wait — actually "ALLOY" has letters A, L, L, O, Y so \(X = \{A, L, O, Y\}\), and "LOYAL" has letters L, O, Y, A, L so \(B = \{L, O, Y, A\}\). Then \(X = B\) since both have the same elements \(\{A, L, O, Y\}\).

(ii) \(A = \{-2, -1, 0, 1, 2\}\). \(B\): solving \(x^2 - 3x + 2 = 0\) gives \(x = 1, 2\), so \(B = \{1, 2\}\). Since \(0 \in A\) and \(0 \notin B\), \(A \neq B\).

1.6 Subsets

Consider the sets \(X = \{\text{set of all students in your school}\}\) and \(Y = \{\text{set of all students in your class}\}\). We note that every element of Y is also an element of X. We say that Y is a subset? of X.

Definition
A set A is said to be a subset of a set B if every element of A is also an element of B. Symbolically, we write \(A \subset B\).

In other words, \(A \subset B\) if whenever \(a \in A\), then \(a \in B\). It is often convenient to use the symbol "\(\Rightarrow\)" which means implies. Using this symbol, we can write the definition of subset as follows:
\[A \subset B \text{ if } a \in A \Rightarrow a \in B\]

If A is not a subset of B, we write \(A \not\subset B\). For A to be a subset of B, all that is needed is that every element of A is in B. It is possible that every element of B may or may not be in A.

If it so happens that every element of B is also in A, then we shall also have \(B \subset A\). In this case, A and B are the same sets so that we have \(A \subset B\) and \(B \subset A \Leftrightarrow A = B\), where "\(\Leftrightarrow\)" is a symbol for two-way implications, read as "if and only if".

Important Results
  • Every set is a subset of itself: \(A \subset A\).
  • The empty set is a subset of every set: \(\emptyset \subset A\) for all sets A.
  • If \(A \subset B\) and \(A \neq B\), then A is called a proper subset of B, and B is called a superset of A. Thus \(\{1, 2, 3\}\) is a proper subset of \(\{1, 2, 3, 4\}\).
  • If a set A has only one element, we call it a singleton set. Thus \(\{a\}\) is a singleton set.

Example 9

Consider the sets \(\emptyset\), \(A = \{1, 3\}\), \(B = \{1, 5, 9\}\), \(C = \{1, 3, 5, 7, 9\}\). Insert the symbol \(\subset\) or \(\not\subset\) between each of the following pair of sets:

(i) \(\emptyset \;\_\;\; B\)   (ii) \(A \;\_\;\; B\)   (iii) \(A \;\_\;\; C\)   (iv) \(B \;\_\;\; C\)

Solution
(i) \(\emptyset \subset B\) as the empty set is a subset of every set.
(ii) \(A \not\subset B\) as \(3 \in A\) but \(3 \notin B\).
(iii) \(A \subset C\) as 1, 3 also belong to C.
(iv) \(B \subset C\) as each element of B is also an element of C.

Example 10

Let \(A = \{a, e, i, o, u\}\) and \(B = \{a, b, c, d\}\). Is A a subset of B? No. (Why?). Is B a subset of A? No. (Why?)

Solution
A is not a subset of B because \(e \in A\) but \(e \notin B\).
B is not a subset of A because \(b \in B\) but \(b \notin A\).
Note that an element of a set can never be a subset of itself.

Example 11

Let A, B and C be three sets. If \(A \in B\) and \(B \subset C\), is it true that \(A \subset C\)? If not, give an example.

Solution
No. Let \(A = \{1\}\), \(B = \{\{1\}, 2\}\) and \(C = \{\{1\}, 2, 3\}\). Here \(A \in B\) as \(\{1\}\) is an element of B and \(B \subset C\). But \(A \not\subset C\) as \(1 \in A\) and \(1 \notin C\). Note that an element of a set can never be a subset of itself.

1.6.1 Intervals as Subsets of R

Let \(a, b \in \mathbb{R}\) and \(a \lt b\). Then the set of real numbers \(\{y : a \lt y \lt b\}\) 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 \(a\) and \(b\) themselves do not belong to this interval.

The interval which contains the end points also is called a closed interval? and is denoted by \([a, b]\). Thus \([a, b] = \{x : a \leq x \leq b\}\).

We can also have intervals closed at one end and open at the other:

  • \([a, b) = \{x : a \leq x \lt b\}\) — open from \(a\) to \(b\), including \(a\) but excluding \(b\)
  • \((a, b] = \{x : a \lt x \leq b\}\) — open from \(a\) to \(b\), excluding \(a\) but including \(b\)
(a, b) a b [a, b] a b [a, b) a b (a, b] a b = open (not included) = closed (included)
Fig 1.1 — Various types of intervals on the real number line

The number \((b - a)\) is called the length of any of the intervals \((a, b)\), \([a, b]\), \([a, b)\) or \((a, b]\).

Important Subsets of R
\(\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}\)
Also, \(\mathbb{T} \subset \mathbb{R}\), where \(\mathbb{T}\) is the set of irrational numbers, i.e., \(\mathbb{T} = \{x : x \in \mathbb{R} \text{ and } x \notin \mathbb{Q}\}\). Members of \(\mathbb{T}\) include \(\sqrt{2}\), \(\sqrt{5}\), and \(\pi\).

1.7 Power Set

Definition
The collection of all subsets of a set A is called the power set? of A. It is denoted by \(P(A)\). In \(P(A)\), every element is a set.

For example, if \(A = \{1, 2\}\), then \(P(A) = \{\emptyset, \{1\}, \{2\}, \{1, 2\}\}\).

Key Result
If A has \(n\) elements, i.e., \(n(A) = n\), then the number of elements in \(P(A)\) is \(2^n\), i.e., \(n[P(A)] = 2^n\).

For example: if \(A = \{1, 2, 3\}\), then \(n(A) = 3\) and \(n[P(A)] = 2^3 = 8\).

Let us verify: \(P(\{1, 2, 3\}) = \{\emptyset, \{1\}, \{2\}, \{3\}, \{1,2\}, \{1,3\}, \{2,3\}, \{1,2,3\}\}\). That is indeed 8 subsets.

1.8 Universal Set

Usually, in a particular context, we have to deal with the elements and subsets of a basic set which is relevant to that particular context. For example, while studying the system of natural numbers and its subsets such as the set of all prime numbers, the set of all even numbers, and so forth, the basic set is the set of natural numbers.

Definition
This basic set is called the universal set?. The universal set is usually denoted by U, and all its subsets by the letters A, B, C, etc.

For example, for the set of all integers, the universal set can be the set of rational numbers or, for that matter, the set \(\mathbb{R}\) of real numbers. For another example, in human population studies, the universal set consists of all the people in the world.

Power Set Generator
Enter a set (comma-separated elements) and see all its subsets

Exercise 1.3

Q1. Make correct statements by filling in the symbols \(\subset\) or \(\not\subset\) in the blank spaces:
(i) \(\{2, 3, 4\} \;\_\;\; \{1, 2, 3, 4, 5\}\)
(ii) \(\{a, b, c\} \;\_\;\; \{b, c, d\}\)
(iii) \(\{x : x \text{ is a student of Class XI of your school}\} \;\_\;\; \{x : x \text{ is a student of your school}\}\)
(iv) \(\{x : x \text{ is a circle in the plane}\} \;\_\;\; \{x : x \text{ is a circle in the same plane with radius 1 unit}\}\)
(v) \(\{x : x \text{ is a triangle in a plane}\} \;\_\;\; \{x : x \text{ is a rectangle in the plane}\}\)
(vi) \(\{x : x \text{ is an equilateral triangle in a plane}\} \;\_\;\; \{x : x \text{ is a triangle in the same plane}\}\)
(vii) \(\{x : x \text{ is an even natural number}\} \;\_\;\; \{x : x \text{ is an integer}\}\)
(i) \(\{2, 3, 4\} \subset \{1, 2, 3, 4, 5\}\)
(ii) \(\{a, b, c\} \not\subset \{b, c, d\}\) (since \(a \notin \{b, c, d\}\))
(iii) \(\subset\) (every Class XI student is a student of the school)
(iv) \(\not\subset\) (a circle of radius 2 is a circle in the plane but not a circle of radius 1)
(v) \(\not\subset\) (a triangle is not a rectangle)
(vi) \(\subset\) (every equilateral triangle is a triangle)
(vii) \(\subset\) (every even natural number is an integer)
Q2. Examine whether the following statements are true or false:
(i) \(\{a, b\} \not\subset \{b, c, a\}\)
(ii) \(\{a, e\} \subset \{x : x \text{ is a vowel in the English alphabet}\}\)
(iii) \(\{1, 2, 3\} \subset \{1, 3, 5\}\)
(iv) \(\{a\} \subset \{a, b, c\}\)
(v) \(\{a\} \in \{a, b, c\}\)
(vi) \(\{x : x \text{ is an even natural number less than } 6\} \subset \{x : x \text{ is a natural number which divides } 36\}\)
(i) False. \(\{a, b\} \subset \{b, c, a\}\) since both \(a\) and \(b\) are in \(\{b, c, a\}\).
(ii) True. \(a\) and \(e\) are vowels.
(iii) False. \(2 \in \{1, 2, 3\}\) but \(2 \notin \{1, 3, 5\}\).
(iv) True. The element \(a\) of \(\{a\}\) is in \(\{a, b, c\}\).
(v) False. \(\{a\}\) is a set; the elements of \(\{a, b, c\}\) are \(a, b, c\) (not \(\{a\}\)).
(vi) True. Even numbers less than 6: \(\{2, 4\}\). Divisors of 36: \(\{1,2,3,4,6,9,12,18,36\}\). Since \(2, 4 \in\) divisors of 36, the statement holds.
Q3. Let \(A = \{1, 2, \{3, 4\}, 5\}\). Which of the following statements are incorrect and why?
(i) \(\{3, 4\} \subset A\)
(ii) \(\{3, 4\} \in A\)
(iii) \(\{\{3, 4\}\} \subset A\)
(iv) \(1 \in A\)
(v) \(1 \subset A\)
(vi) \(\{1, 2, 5\} \subset A\)
(vii) \(\{1, 2, 5\} \in A\)
(viii) \(\{1, 2, 3\} \subset A\)
(ix) \(\emptyset \in A\)
(x) \(\emptyset \subset A\)
(xi) \(\{\emptyset\} \subset A\)
(i) Incorrect. 3 is not an element of A (the element of A is the set \(\{3, 4\}\), not 3 alone).
(ii) Correct. \(\{3, 4\}\) is indeed an element of A.
(iii) Correct. \(\{\{3, 4\}\} \subset A\) because its sole element \(\{3, 4\}\) is in A.
(iv) Correct. 1 is an element of A.
(v) Incorrect. 1 is not a set, so the subset relation is not applicable.
(vi) Correct. 1, 2, and 5 are all elements of A.
(vii) Incorrect. \(\{1, 2, 5\}\) is not an element of A.
(viii) Incorrect. 3 is not an element of A.
(ix) Incorrect. \(\emptyset\) is not an element of A.
(x) Correct. \(\emptyset\) is a subset of every set.
(xi) Incorrect. \(\emptyset\) is not an element of A, so \(\{\emptyset\}\) is not a subset.
Q4. Write down all the subsets of the following sets:
(i) \(\{a\}\)   (ii) \(\{a, b\}\)   (iii) \(\{1, 2, 3\}\)   (iv) \(\emptyset\)
(i) Subsets of \(\{a\}\): \(\emptyset, \{a\}\).
(ii) Subsets of \(\{a, b\}\): \(\emptyset, \{a\}, \{b\}, \{a, b\}\).
(iii) Subsets of \(\{1, 2, 3\}\): \(\emptyset, \{1\}, \{2\}, \{3\}, \{1,2\}, \{1,3\}, \{2,3\}, \{1,2,3\}\).
(iv) Subsets of \(\emptyset\): \(\emptyset\). (The only subset of the empty set is itself.)
Q5. Write the following as intervals:
(i) \(\{x : x \in \mathbb{R}, -4 \lt x \leq 6\}\)
(ii) \(\{x : x \in \mathbb{R}, -12 \lt x \lt -10\}\)
(iii) \(\{x : x \in \mathbb{R}, 0 \leq x \lt 7\}\)
(iv) \(\{x : x \in \mathbb{R}, 3 \leq x \leq 4\}\)
(i) \((-4, 6]\)
(ii) \((-12, -10)\)
(iii) \([0, 7)\)
(iv) \([3, 4]\)
Q6. Write the following intervals in set-builder form:
(i) \((-3, 0)\)   (ii) \([6, 12]\)   (iii) \((6, 12]\)   (iv) \([-23, 5)\)
(i) \(\{x : x \in \mathbb{R}, -3 \lt x \lt 0\}\)
(ii) \(\{x : x \in \mathbb{R}, 6 \leq x \leq 12\}\)
(iii) \(\{x : x \in \mathbb{R}, 6 \lt x \leq 12\}\)
(iv) \(\{x : x \in \mathbb{R}, -23 \leq x \lt 5\}\)
Q7. What universal set(s) would you propose for each of the following:
(i) The set of right triangles.   (ii) The set of isosceles triangles.
(i) The set of all triangles in a plane (or the set of all polygons).
(ii) The set of all triangles in a plane.
Q8. Given the sets \(A = \{1, 3, 5\}\), \(B = \{2, 4, 6\}\) and \(C = \{0, 2, 4, 6, 8\}\), which of the following may be considered as universal set(s) for all the three sets A, B and C?
(i) \(\{0, 1, 2, 3, 4, 5, 6\}\)
(ii) \(\emptyset\)
(iii) \(\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\)
(iv) \(\{1, 2, 3, 4, 5, 6, 7, 8\}\)
A universal set must contain all elements of A, B and C. The union \(A \cup B \cup C = \{0, 1, 2, 3, 4, 5, 6, 8\}\).
(i) No. Does not contain 8.
(ii) No. Empty set contains no elements at all.
(iii) Yes. Contains all elements 0, 1, 2, 3, 4, 5, 6, 8 (and more).
(iv) No. Does not contain 0.
Activity: Exploring Subsets and Power Sets
Materials: Pen and paper
Predict: If a set has 4 elements, how many subsets do you think it will have? Make your guess before proceeding.
  1. Start with the empty set \(\emptyset\). List its subsets. How many? (Answer: 1, which is \(2^0\))
  2. Take \(\{a\}\). List all subsets: \(\emptyset, \{a\}\). Count = 2 = \(2^1\).
  3. Take \(\{a, b\}\). List all subsets: \(\emptyset, \{a\}, \{b\}, \{a, b\}\). Count = 4 = \(2^2\).
  4. Take \(\{a, b, c\}\). List all 8 subsets. Verify = \(2^3\).
  5. Now try \(\{a, b, c, d\}\). Can you list all 16 subsets?
  6. Observe the pattern: each new element doubles the number of subsets (because each existing subset spawns two: one with and one without the new element).

Observe: For \(\{a, b, c, d\}\), the 16 subsets are: \(\emptyset\), \(\{a\}\), \(\{b\}\), \(\{c\}\), \(\{d\}\), \(\{a,b\}\), \(\{a,c\}\), \(\{a,d\}\), \(\{b,c\}\), \(\{b,d\}\), \(\{c,d\}\), \(\{a,b,c\}\), \(\{a,b,d\}\), \(\{a,c,d\}\), \(\{b,c,d\}\), \(\{a,b,c,d\}\).

Explain: The formula \(2^n\) works because for each element, you have 2 choices: include it or exclude it. With \(n\) elements, total choices = \(2 \times 2 \times \ldots \times 2 = 2^n\).

Competency-Based Questions
A sports academy has 120 athletes. Let U be the set of all athletes. The coaching staff classifies them into sets: S (swimmers, 45 members), R (runners, 60 members), G (gymnasts, 30 members). Some athletes participate in multiple sports. The administrator uses set theory to organise training schedules and facility bookings.
Q1. The administrator needs to book the swimming pool for all subsets of the swim team of size 5 (for relay practice). If the swim team has 6 members in a particular relay squad, how many different 5-person relay teams can be formed? Express your working using the concept of subsets.
L3 Apply
We need to count the number of 5-element subsets of a 6-element set. Using combinations: \(\binom{6}{5} = 6\). So 6 different relay teams can be formed. Note: while the power set has \(2^6 = 64\) total subsets, we only need those of size 5.
Q2. Analyse the relationship between S, R, and G. Is it possible that \(S \subset R\)? Under what conditions would \(S = R\)? What does \(n(S) \lt n(R)\) tell us about the subset relationship between S and R?
L4 Analyse
\(S \subset R\) would mean every swimmer is also a runner. This is possible but unlikely in practice. \(S = R\) would require every swimmer to be a runner AND every runner to be a swimmer, meaning both sets have identical members. Since \(n(S) = 45 \neq 60 = n(R)\), we know \(S \neq R\). However, \(n(S) \lt n(R)\) alone does NOT guarantee \(S \subset R\): S could contain members not in R. The cardinality comparison only tells us S cannot be a superset of R.
Q3. A coach claims: "Since the empty set is a subset of every set, an athlete who plays no sport is automatically a member of the swimming team." Evaluate whether this reasoning is mathematically valid.
L5 Evaluate
The reasoning is invalid. The statement \(\emptyset \subset S\) means that the empty set (as a set) is a subset of S — NOT that "nothing" is an element of S. Being a subset is a relationship between sets; being a member is a relationship between an object and a set. An athlete who plays no sport is not represented by \(\emptyset\) — they are simply a person \(p\) where \(p \notin S\). The coach is confusing the subset relation (\(\subset\)) with the membership relation (\(\in\)).
Q4. Design a classification system for the academy where every athlete belongs to exactly one primary training group. Create at least 4 non-overlapping subsets of U, define them using set-builder notation, and verify that their union equals U.
L6 Create
One possible system:
\(A_1 = \{x \in U : x \text{ is primarily a swimmer}\}\) (assigned based on most hours)
\(A_2 = \{x \in U : x \text{ is primarily a runner}\}\)
\(A_3 = \{x \in U : x \text{ is primarily a gymnast}\}\)
\(A_4 = \{x \in U : x \text{ participates in other/no sports}\}\)
By assigning each athlete exactly one primary sport based on their most-trained discipline, we ensure: \(A_i \cap A_j = \emptyset\) for \(i \neq j\) (disjoint), and \(A_1 \cup A_2 \cup A_3 \cup A_4 = U\) (exhaustive). Every athlete falls into exactly one category.

Assertion–Reason Questions

Assertion (A): If \(A = \{1, 2\}\), then the power set \(P(A)\) has 4 elements.
Reason (R): If a set has \(n\) elements, its power set has \(2^n\) elements.
(a) Both A and R are true, and R is the correct explanation of A.
(b) Both A and R are true, but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (a) — \(A\) has 2 elements, so \(|P(A)| = 2^2 = 4\). Both true, and R directly explains A.
Assertion (A): The interval \((3, 5]\) contains the number 3.
Reason (R): In an interval \((a, b]\), the left endpoint is excluded and the right endpoint is included.
(a) Both A and R are true, and R is the correct explanation of A.
(b) Both A and R are true, but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (d) — A is false: the round bracket at 3 means 3 is excluded from \((3, 5]\). R is true and correctly explains why A is false.
Assertion (A): \(\{1, 2\} \subset \{1, 2, 3\}\) and \(\{1, 2\} \in P(\{1, 2, 3\})\).
Reason (R): Every subset of a set A is an element of the power set \(P(A)\).
(a) Both A and R are true, and R is the correct explanation of A.
(b) Both A and R are true, but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (a) — Both are true. Since \(\{1,2\}\) is a subset of \(\{1,2,3\}\), it must appear as an element in the power set. R is the exact reason why A holds.

Frequently Asked Questions

What is a subset in set theory?

A set A is a subset of set B if every element of A is also an element of B. Every set is a subset of itself, and the empty set is a subset of every set.

What is a power set?

The power set of a set A is the collection of all subsets of A, denoted P(A). If A has n elements, then P(A) has 2^n elements.

What are intervals on the number line?

Intervals are subsets of real numbers between two endpoints. Open interval (a,b) excludes endpoints, closed interval [a,b] includes them, and half-open intervals include one endpoint.

What is the difference between subset and proper subset?

If A is a subset of B and A is not equal to B, then A is called a proper subset of B. Every set is a subset of itself but not a proper subset of itself.

How do you find the number of subsets of a set?

A set with n elements has exactly 2^n subsets, including the empty set and the set itself. For example, a set with 3 elements has 8 subsets.

Frequently Asked Questions — Sets

What is Subsets, Intervals, and Power Sets in NCERT Class 11 Mathematics?

Subsets, Intervals, and Power Sets is a key concept covered in NCERT Class 11 Mathematics, Chapter 1: Sets. This lesson builds the student's foundation in the chapter by explaining the core ideas with worked examples, definitions, and step-by-step methods aligned to the CBSE curriculum.

How do I solve problems on Subsets, Intervals, and Power Sets step by step?

To solve problems on Subsets, Intervals, and Power Sets, follow the NCERT method: identify the given quantities, choose the relevant formula or theorem, substitute values carefully, and simplify. Class 11 exercises gradually increase in difficulty — start with solved NCERT examples before attempting exercise questions, and always verify your answer by substitution or diagram.

What are the most important formulas for Chapter 1: Sets?

The essential formulas of Chapter 1 (Sets) are listed in the chapter summary and highlighted throughout the lesson in formula boxes. Memorise them and practise at least 2–3 problems per formula. CBSE board exams frequently test direct application as well as combined use of multiple formulas from this chapter.

Is Subsets, Intervals, and Power Sets important for the Class 11 board exam?

Subsets, Intervals, and Power Sets is part of the NCERT Class 11 Mathematics syllabus and appears in CBSE board exams. Questions typically include short-answer, long-answer, and competency-based items. Review the NCERT examples, exercise questions, and previous-year board problems on this topic to prepare confidently.

What mistakes should students avoid in Subsets, Intervals, and Power Sets?

Common mistakes in Subsets, Intervals, and Power Sets include skipping steps, misapplying formulas, sign errors, and losing track of units. Write each step clearly, double-check algebraic manipulations, and re-read the question after solving to verify that your answer matches what was asked.

Where can I find more NCERT practice questions on Subsets, Intervals, and Power Sets?

End-of-chapter NCERT exercises for Subsets, Intervals, and Power Sets cover all difficulty levels tested in CBSE exams. After completing them, try the examples again without looking at the solutions, attempt the NCERT Exemplar questions for Chapter 1, and solve at least one previous-year board paper to consolidate your understanding.

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