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Sets and Functions - Sets, their representations, and types

Grade 11ICSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A Set is a well-defined collection of distinct objects. Objects in a set are called elements or members. We use capital letters (e.g., A,B,SA, B, S) to denote sets and small letters (e.g., a,b,xa, b, x) for elements. If xx is an element of set AA, we write x∈Ax \in A.

Venn diagram showing element x inside set A and element y outside set A.
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Sets can be represented in two ways: Roster (Tabular) Form, where elements are listed within braces separated by commas, e.g., V={a,e,i,o,u}V = \{a, e, i, o, u\}; and Set-Builder Form, where we state the common property of elements, e.g., V={x:x is a vowel in English alphabet}V = \{x : x \text{ is a vowel in English alphabet}\}.

Diagram
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A Subset A⊆BA \subseteq B means every element of AA is also in BB. If there is at least one element in BB not in AA, then AA is a Proper Subset (A⊂BA \subset B). An Empty Set (∅\emptyset or {}\{\}) has no elements and is a subset of every set.

Venn diagram showing circle A completely inside circle B representing A as a subset of B.
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Types of Sets include: Finite Sets (countable elements), Infinite Sets (uncountable, e.g., N\mathbb{N}), Singleton Sets (exactly one element), and Equal Sets (identical elements regardless of order).

Visual representation of a Singleton set containing one element and a Null set containing no elements.

📐Formulae

Cardinal number of a set AA: n(A)n(A)

Number of subsets of a set with nn elements: 2n2^n

Number of proper subsets of a set with nn elements: 2n−12^n - 1

Condition for Equality: A=B  ⟺  A⊆B and B⊆AA = B \iff A \subseteq B \text{ and } B \subseteq A

Power Set notation: P(A)={S:S⊆A}P(A) = \{S : S \subseteq A\}

Set-Builder notation: {x∣P(x)} where P(x) is the property satisfied by x\{x \mid P(x)\} \text{ where } P(x) \text{ is the property satisfied by } x

💡Examples

Problem 1:

Write the set A={x:x∈Z,x2<20}A = \{x : x \in \mathbb{Z}, x^2 < 20\} in Roster form and find its cardinal number n(A)n(A).

Solution:

  1. Identify the condition: xx must be an integer (Z\mathbb{Z}) and its square must be less than 2020.
  2. Test integers: 02=0<200^2 = 0 < 20 (Yes) (±1)2=1<20(\pm 1)^2 = 1 < 20 (Yes) (±2)2=4<20(\pm 2)^2 = 4 < 20 (Yes) (±3)2=9<20(\pm 3)^2 = 9 < 20 (Yes) (±4)2=16<20(\pm 4)^2 = 16 < 20 (Yes) (±5)2=25≮20(\pm 5)^2 = 25 \not< 20 (No)
  3. List the elements: A={−4,−3,−2,−1,0,1,2,3,4}A = \{-4, -3, -2, -1, 0, 1, 2, 3, 4\}.
  4. Count the elements: There are 99 elements in total.
  5. Therefore, n(A)=9n(A) = 9.

Explanation:

This problem requires converting a logic-based set-builder notation into a specific list of elements by checking which integers satisfy the inequality x2<20x^2 < 20.

Problem 2:

If S={1,2,3}S = \{1, 2, 3\}, find the Power Set P(S)P(S) and verify the number of elements.

Solution:

  1. List all possible subsets of SS:
    • Subsets with 0 elements: ϕ\phi
    • Subsets with 1 element: {1},{2},{3}\{1\}, \{2\}, \{3\}
    • Subsets with 2 elements: {1,2},{1,3},{2,3}\{1, 2\}, \{1, 3\}, \{2, 3\}
    • Subsets with 3 elements: {1,2,3}\{1, 2, 3\}
  2. Combine them into one set: P(S)={ϕ,{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}P(S) = \{\phi, \{1\}, \{2\}, \{3\}, \{1, 2\}, \{1, 3\}, \{2, 3\}, \{1, 2, 3\}\}.
  3. Verify the count: The number of elements in SS is n=3n = 3. The formula for the number of elements in P(S)P(S) is 2n=23=82^n = 2^3 = 8. Counting the listed subsets, we get exactly 88.

Explanation:

The Power Set is the set of all subsets. Systematic listing (by number of elements) ensures no subset is missed, and the 2n2^n formula serves as a check for accuracy.

Problem 3:

Given the Universal set U={1,2,3,4,5,6,7,8}U = \{1, 2, 3, 4, 5, 6, 7, 8\} and set A={x:x∈U and x is even}A = \{x : x \in U \text{ and } x \text{ is even}\}, represent AA in Roster form and visualize its position in UU.

A Venn diagram where set A containing even numbers 2, 4, 6, 8 is inside a rectangle U containing odd numbers 1, 3, 5, 7 outside the circle.

Solution:

In UU, the even numbers are 2,4,6,82, 4, 6, 8. Therefore, in Roster form: A={2,4,6,8}A = \{2, 4, 6, 8\}. The cardinal number n(A)=4n(A) = 4.

Explanation:

We filter the elements of UU based on the condition 'is even' to form set AA.

Problem 4:

Given the Universal set U={1,2,3,4,5,6,7,8,9,10}U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} and two sets A={2,4,6,8,10}A = \{2, 4, 6, 8, 10\} and B={1,3,5,7,9}B = \{1, 3, 5, 7, 9\}, identify the relationship between AA and BB. Represent these sets using a Venn diagram and determine if they are disjoint.

Venn diagram showing two disjoint sets A and B within a universal set U. Set A contains even numbers and set B contains odd numbers.

Solution:

U={1,2,3,4,5,6,7,8,9,10}U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} A={2,4,6,8,10}A = \{2, 4, 6, 8, 10\} B={1,3,5,7,9}B = \{1, 3, 5, 7, 9\} Checking for common elements: A∩B=∅A \cap B = \emptyset Since there are no common elements between set AA and set BB, they are called Disjoint Sets. Cardinality: n(A)=5n(A) = 5 n(B)=5n(B) = 5 Since n(A)=n(B)n(A) = n(B), they are also Equivalent Sets.

Explanation:

Two sets are disjoint if their intersection is an empty set. Here, AA consists of even numbers and BB consists of odd numbers from the universal set UU. Since no number is both even and odd, the sets do not overlap. The Venn diagram shows two separate circles within the rectangle representing the Universal set.

Sets, their representations, and types Class 11 Notes & Examples