Business Mathematics and Basic Statistics · Class 11 Commerce
Ch 9Sets — Operations and Functions — Class 11 Business Mathematics and Basic Statistics, concept-first.
You already know from the previous chapter that a set is a subset of a set (written ) whenever every element of is also an element of . The empty set deserves special attention here, because a genuinely important — and, at first glance, slightly strange-looking — fact follows directly from this definition.
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
Power Set and Cardinality
For a set , the power set is the set of ALL subsets of , always including (since for every set ) and itself. For a finite set with elements, , because each element independently is either included in, or excluded from, a…
Most relevant Q&A
- Let $A = \{x, y\}$. Write the power set $P(A)$ and verify that $n(P(A)) = 2^2$.Free
- Let $A = \{p, q, r\}$. Verify that $\emptyset \subseteq A$, then write the power set $P(A)$ and state $n(P(A))$.Free
- Let $A = \{1, 2, 3, 4\}$. Write the power set $P(A)$ and verify that $n(P(A)) = 2^4$ by counting the subsets of each possible size separatel…Free
In previous exams
How often this chapter’s concepts have been examined — real appearance data, never estimated.
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Revisiting the Null Set — A Subset of Every Set
You already know from the previous chapter that a set is a subset of a set (written ) whenever every element of is also an element of .
Power Set of a Finite Set and Its Cardinality
Given a set , the power set of — written — is the set of all possible subsets of , including itself and the empty set . In symbols,
Set Operations — Union and Intersection
Just as numbers can be combined using addition and multiplication, sets can be combined using set operations. The two most basic ones are union and intersection.
Set Operations — Difference, Symmetric Difference and Complement
Three more operations round out the standard set-operation toolkit.
Disjoint Sets
Two sets and are called disjoint if they share no elements at all — that is, their intersection is empty:
Venn Diagrams — Picturing Set Operations
A Venn diagram represents sets as overlapping regions, usually drawn as circles (or ovals) inside a rectangle representing the universal set .
Commutative and Distributive Properties of Set Operations
Union and intersection both satisfy a commutative property — the order in which the two sets are combined does not matter: Drawing the Venn diagram for and for produces the exact same shaded region (b…
De Morgan's Laws
De Morgan's Laws describe how the complement of a union or intersection relates to the complements of the individual sets — they are among the most useful identities in this chapter, and come up const…
The Inclusion-Exclusion Principle — Two Sets
Section 3 already hinted at a problem: simply adding over-counts any elements shared between and , because those elements get counted once inside and again inside .
The Inclusion-Exclusion Principle — Three Sets
The same idea extends to three overlapping sets , , , though the bookkeeping is a little more involved: simply adding all three individual counts and subtracting all three pairwise overlaps ends up su…
Cartesian Product and Ordered Pairs — A Quick Recap
The previous chapter introduced the Cartesian product of two finite sets and : the set of all ordered pairs formed by taking one element from (first) and one from (second), with .
Functions — Pictorial Representation, Domain, Co-domain and Range
A function from a set to a set is a rule that assigns to every element of exactly one element of . This syllabus studies functions entirely through their pictorial representation — an arrow diagram —…
Classifying Functions — One-One, Many-One, Onto and Bijective
Once a diagram is confirmed to represent a function, it can be classified further by how its arrows are arranged.
Exercises
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- Q12Let $A = \{x, y\}$. Write the power set $P(A)$ and verify that $n(P(A)) = 2^2$.Free
- Q13Let $A = \{1, 2, 3, 4\}$, $B = \{2, 3, 5\}$ and $C = \{3, 4, 6\}$. Verify the distributive property $A \cap (B \cup C) = (A \cap B) \cup (A…Free
- Q14Let $U = \{1, 2, \dots, 10\}$, $A = \{1,3,5,7,9\}$ and $B = \{2,3,5,7\}$. Verify De Morgan's Law $(A \cap B)' = A' \cup B'$.Free
- Q15In a group of 60 people, 35 like tea, 30 like coffee, and 40 like at least one of the two drinks. Find how many people like both tea and cof…Preview
- Q16Among 120 students, 60 study Accountancy, 50 study Business Mathematics, and 45 study Costing. Of these, 25 study both Accountancy and Busin…Preview
- Q17Let $A = \{a, b, c\}$ and $B = \{1, 2, 3, 4\}$, with the arrows $a \to 2$, $b \to 4$, $c \to 2$. Is this a function from $A$ to $B$? If so,…Preview
- Q18(i) Let $A = \{1,2,3,4\}$ and $B = \{5,6\}$, with arrows $1\to5$, $2\to5$, $3\to6$, $4\to6$. Classify this function. (ii) Let $A = \{1,2,3\}…Preview
More questions
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- Example 1Let $A = \{p, q, r\}$. Verify that $\emptyset \subseteq A$, then write the power set $P(A)$ and state $n(P(A))$.Free
- Example 2Let $A = \{1, 2, 3, 4\}$. Write the power set $P(A)$ and verify that $n(P(A)) = 2^4$ by counting the subsets of each possible size separatel…Free
- Example 3Let $A = \{2, 4, 6, 8, 10\}$ and $B = \{4, 8, 12, 16\}$. Find $A \cup B$ and $A \cap B$. Then verify that $n(A \cup B) = n(A) + n(B) - n(A \…Free
- Example 4Let $U = \{1, 2, \dots, 10\}$, $A = \{1,2,3,4,5\}$ and $B = \{4,5,6,7,8\}$. Find: (i) $A - B$ (ii) $B - A$ (iii) $A \triangle B$ (iv) $A'$ a…Preview
- Example 5Let $A = \{a, b, c, d\}$ and $B = \{c, d, e, f\}$. Verify the commutative properties $A \cup B = B \cup A$ and $A \cap B = B \cap A$ by list…Preview
- Example 6Let $A = \{1, 2, 3\}$, $B = \{2, 3, 4\}$ and $C = \{3, 4, 5\}$. Verify the distributive property $A \cup (B \cap C) = (A \cup B) \cap (A \cu…Preview
- Example 7Let $U = \{1, 2, \dots, 10\}$, $A = \{1,2,3,4\}$ and $B = \{3,4,5,6\}$. Verify De Morgan's Law $(A \cup B)' = A' \cap B'$.Preview
- Example 8In a class of 50 students, 30 play cricket, 25 play football, and 12 play both games. Find how many students play at least one of the two ga…Preview
- Example 9In a survey of 100 people, 45 read newspaper A, 40 read newspaper B, and 38 read newspaper C. Of these, 20 read both A and B, 15 read both B…Preview
- Example 10Let $A = \{1, 2, 3, 4\}$ and $B = \{p, q, r, s, t\}$. Diagram (i) shows the arrows $1\to p$, $2\to q$, $3\to r$, $4\to q$. Diagram (ii) show…Preview
- Example 11Let $A = \{1, 2, 3\}$ and $B = \{4, 5, 6\}$, with the arrows $1\to 4$, $2\to 5$, $3\to 6$. Classify this function as one-one or many-one, an…Preview