Business Mathematics and Statistics · Class 12 Commerce
Ch 3Set Theory — Class 12 Business Mathematics and Statistics, concept-first.
A set is simply a well-defined collection of distinct objects — well-defined meaning that for any given object we can decide, with total certainty, whether it belongs to the collection or not.
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
Meaning, Notation and Representation of a Set
A set is a well-defined collection of distinct objects, with membership shown by and non-membership by . It can always be written in either of two equivalent forms: roster form, which lists every element explicitly (e.g.…
Most relevant Q&A
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Meaning and Notation of a Set
A set is simply a well-defined collection of distinct objects — well-defined meaning that for any given object we can decide, with total certainty, whether it belongs to the collection or not.
Representing a Set — Roster Form and Set-Builder Form
A set can be written down in two standard, fully interchangeable ways.
Types of Sets I — Empty, Finite, Infinite, Singleton and Universal Sets
A handful of set types recur often enough to deserve their own names.
Types of Sets II — Equal Sets, Equivalent Sets, Subsets, Proper Subset and Power Set
Beyond the special sets above, several relationships between sets matter just as much.
Set Operations — Union, Intersection, Difference and Complement
Given sets and drawn from a universal set , four operations combine or compare them.
Laws of Set Algebra
Set operations obey algebraic laws closely parallel to ordinary arithmetic, and every one of them can be checked directly against the definitions above (or read off a Venn diagram).
Cardinality and the Inclusion-Exclusion Principle
The cardinality of a finite set , written , is simply the count of its distinct elements. ; if is a singleton, .
Cartesian Product of Sets
Given two non-empty sets and , the Cartesian product collects every possible ordered pair with its first entry from and second entry from :
Exercises
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- Q8Write the set $\{7, 14, 21, 28, 35\}$ in set-builder form.Free
- Q9Classify each of the following as empty, singleton, finite or infinite: (i) $\{x : x \in \mathbb{N}, x + 4 = 2\}$ (ii) $\{x : x \in \mathbb{…Free
- Q10Which of the following pairs of sets are equal? (a) $\{1,2,3\}$ and $\{3,2,1\}$ (b) $\{1,2,3\}$ and $\{1,2,3,4\}$ (c) $\{a,b\}$ and $\{a,a,b…Free
- Q11If $A = \{a,e,i,o,u\}$ and $B = \{a,b,c,d,e\}$, find $A \cup B$, $A \cap B$ and $A - B$.Preview
- Q12Verify the distributive law $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ using $A=\{1,2,3,4\}$, $B=\{3,4,5\}$, $C=\{4,5,6\}$.Preview
- Q13In a survey of 150 small retailers in an Odisha district, 80 stock stationery items, 65 stock grocery items, and 30 stock both. Find (i) how…Preview
- Q14A garment shop stocks shirts in sizes $A=\{S,M,L\}$ and colours $B=\{\text{White},\text{Blue}\}$. List $A \times B$ and state how many disti…Preview
Sample & Board Papers
Sample papers and previous-year board questions for this subject.
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- Q1Express each of the following in one word/term each: (vi) A set which does not contain any element.Preview
- Q2(e) Number of subsets, a set of 4 elements can have, is (a) 8 (b) 16 (c) 32 (d) 64Preview
- Q3(e) If $A = \{1, 3, 5, 7, 9\}$, $B = \{1, 7, 8\}$ and $C = \{3, 5, 8, 10, 12\}$, then find $(A \cup B) \cap (B \cup C)$.Preview
- Q4If set $A = \{11, 12, 13\}$ and set $B = \{12, 13, 14\}$, then the number of elements in set $A \cup B$ is : (a) $4$ (b) $5$ (c) $6$ (d) $7$Preview
- Q5From the following the one which is a null set, is : (a) $\{0\}$ (b) $\{\ \}$ (c) $\{1\}$ (d) $\{1, 2, 3\}$Preview
- Q6The maximum number of subsets that can be formed out of the set $\{2, 3, 4, 5, 6\}$ is : (a) $32$ (b) $8$ (c) $64$ (d) $16$Preview
- Q7What is a set ?Preview
- Q8Show that $A = \{-2, -3\}$ and $B = \{x : x \text{ is the solution of } x^2 + 5x + 6 = 0\}$ are equal sets.Preview
- Q9Write all the elements of the set, $A = \{x : x \text{ is an integer and } x^2 \leq 4\}$.Preview
- Q10Number of proper subsets of a set having 5 elements are : (a) 16 (b) 30 (c) 31 (d) 32Preview
- Q11Express each of the following in one word / term : The set of all elements under study is known asPreview
- Q12Answer the following questions within one sentence each : What is symmetric difference of sets ?Preview
- Q13What do you mean by equivalent sets ?Preview
- Q14If $A = \{a, b, c, d\}$, $B = \{b, c, d, e\}$, prove that : $A - B = (A \cup B) - B$Preview
- Q15When no element of set A is in set B and no element of set B is in set A, the set A and set B are called : (a) Empty sets (b) Disjoint sets…Preview
- Q16Express each of the following in one word / term : A set which contains no elements.Preview
- Q17Write any two subsets of the set, $\{1, 2, 3\}$.Preview
- Q18If $A = \{1, 2, 3, 4\}$, $B = \{2, 4, 6, 8\}$ and $C = \{1, 2, 3, 5\}$; find $(A \cap B) \cap C$.Preview
More questions
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- Example 1Write the set $A = \{5, 10, 15, 20\}$ in set-builder form.Free
- Example 2If $U = \{1, 2, 3, \dots, 10\}$ and $A = \{1, 3, 5, 7, 9\}$, find the complement $A'$.Free
- Example 3Let $A = \{2, 4, 6, 8\}$ and $B = \{4, 8, 12, 16\}$. Find $A \cup B$, $A \cap B$, $A - B$ and $B - A$.Free
- Example 4If a set $A$ has 4 elements, the number of elements in its power set $P(A)$ is: (a) 8 (b) 12 (c) 16 (d) 4Preview
- Example 5Verify De Morgan's law $(A \cup B)' = A' \cap B'$ for $U = \{1,2,\dots,10\}$, $A = \{1,2,3,4,5\}$ and $B = \{4,5,6,7\}$.Preview
- Example 6In a market survey of 120 customers of an Odisha handicrafts store, 70 customers purchase textile products, 50 purchase pottery products, an…Preview
- Example 7In a market survey of 200 shoppers in Bhubaneswar, 90 use UPI, 70 use debit cards, and 60 use cash. Of these, 30 use both UPI and debit card…Preview