Sets Ex 1.3 Class 11 NCERT — Concepts, Proofs, & Solutions

Core Concepts of Exercise 1.3: Subsets & Intervals

To master the sets ex 1 3 class 11 ncert curriculum, we must start with the formal definition of a subset. A set $A$ is said to be a subset of a set $B$ if every element of $A$ is also an element of $B$. We express this relationship mathematically as $A \\subset B$. Symbolically, we write:

$A \\subset B \\iff (a \\in A \\implies a \\in B)$

If $A$ is a subset of $B$, but $A \\neq B$, then $A$ is called a proper subset of $B$, and $B$ is called the superset of $A$. Another core concept of Exercise 1.3 is expressing intervals of real numbers as subsets of $\\mathbb{R}$. Unlike discrete sets, real number sets are continuous. An open interval $(a, b)$ contains all real numbers strictly between $a$ and $b$, while a closed interval $[a, b]$ includes the boundary numbers $a$ and $b$ as well. Finally, the collection of all possible subsets of a set $A$ is called its Power Set, denoted by $P(A)$. If set $A$ has $n$ elements, the power set will have exactly $2^n$ elements.

Key Terminology and Definitions

Subset (⊂)

Set A is a subset of set B if every element of A is also an element of B.

Proper Subset

If A is a subset of B and A is not equal to B (A ≠ B), then A is called a proper subset of B.

Power Set P(A)

The set of all possible subsets of a given set A. If n(A) = m, then n(P(A)) = 2^m.

Universal Set (U)

A context-specific set that contains all possible elements under discussion, from which all other sets are formed.

How to Find and List All Subsets of a Given Set

1. Step 1: Apply the 2^n rule — Determine the number of elements, n, in the set. Use the formula 2^n to calculate how many subsets you must write down.
2. Step 2: Start with the empty set — Always write the empty set (denoted by ∅) first, as it is a subset of every set.
3. Step 3: List single-element subsets — Create separate subsets containing exactly one element from the parent set.
4. Step 4: List multi-element subsets — Progressively combine elements to list subsets of size 2, size 3, and so on, up to size n-1.
5. Step 5: Write the set itself — Conclude by writing the complete set itself, as every set is a subset of itself.

Step-by-Step Solved Examples

• Example 1: Write down all subsets of set A = {x, y, z}. - Step 1: Identify the number of elements. Here, n = 3. - Step 2: Compute total subsets = 2^3 = 8. - Step 3: Write the empty set: ∅. - Step 4: Write singletons: {x}, {y}, {z}. - Step 5: Write doublets: {x, y}, {y, z}, {x, z}. - Step 6: Write the set itself: {x, y, z}. - Final answer: The subsets are ∅, {x}, {y}, {z}, {x, y}, {y, z}, {x, z}, and {x, y, z}.
• Example 2: Write the interval [2, 7) in set-builder form. - Step 1: Analyze the brackets. '[' on the left means 2 is included in the interval. - Step 2: ')' on the right means 7 is excluded from the interval. - Step 3: Use inequality notation: 2 ≤ x < 7. - Step 4: Specify that x belongs to the set of real numbers (R). - Final answer: {x : x ∈ R, 2 ≤ x < 7}.

Board Exam Trap: Element vs. Subset Notation

One of the most common mistakes made in cbse class 11 sets ex 1 3 is confusing the symbol for 'is an element of' ($\\in$) with 'is a subset of' ($\\subset$).

• Let $A = \\{1, 2, \\{3, 4\\}\\}$.
• Here, the elements of $A$ are $1$, $2$, and the set $\\{3, 4\\}$ itself.
• Therefore, $\\{3, 4\\} \\in A$ is True because the set is a direct element.
• However, $\\{3, 4\\} \\subset A$ is False because $3$ and $4$ are not individual elements of $A$.
• To write it as a subset, you must wrap the element inside another layer of set braces: $\\{\\{3, 4\\}\\} \\subset A$ is True.

Practice Questions with Solutions

• Q: Examine whether the statement is true or false: {a} ∈ {a, b, c}. A: Step 1: Identify the symbol used. The symbol is ∈, which stands for 'is an element of'. Step 2: Check the elements of the set on the right-hand side. The elements are 'a', 'b', and 'c'. Step 3: Check if the left-hand side '{a}' is one of these elements. The element is 'a', not '{a}'. Step 4: Therefore, '{a}' is a subset (⊂), not an element (∈). Final answer: The statement is False.
• Q: Write the following interval in interval notation: {x : x ∈ R, -4 < x ≤ 6}. A: Step 1: Look at the left inequality: x > -4. Since it does not include -4, use an open bracket '('. Step 2: Look at the right inequality: x ≤ 6. Since it includes 6, use a closed bracket ']'. Step 3: Combine both endpoints with their respective brackets. Final answer: (-4, 6]
• Q: If a set A has no elements (A = ∅), find the number of elements in its Power Set P(A). A: Step 1: Identify the number of elements in set A. Since A is empty, n(A) = 0. Step 2: Apply the formula for the number of elements in a power set: n(P(A)) = 2^n. Step 3: Substitute n = 0 into the formula: 2^0 = 1. Step 4: The only subset of ∅ is ∅ itself. Hence, P(A) = {∅}. Final answer: 1
• Q: Let A = {1, 2}, B = {1, 2, 3}, and C = {1, 2, 3, 4}. Is it true that A ⊂ B and B ⊂ C? If yes, what is the relation between A and C? A: Step 1: Check if A ⊂ B. Every element of A (1, 2) is in B. Yes, A ⊂ B. Step 2: Check if B ⊂ C. Every element of B (1, 2, 3) is in C. Yes, B ⊂ C. Step 3: By transitive property of subsets, if A is contained in B, and B is contained in C, then A must be contained in C. Step 4: Let's verify: every element of A (1, 2) is indeed in C. Final answer: Yes, and A ⊂ C.

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