CLASS 11 MATHEMATICS
Complete Coursebook • Concepts • Formulae • Solved Examples • Practice
ISC-oriented study guide
A structured course for building concepts, solving standard problems, and preparing for examinations.
How to use this book
Study the concept notes first, then cover the worked examples without looking at the solution. Attempt the practice set
under time pressure. Keep a formula sheet and an error log. Revisit questions you could not solve after 24–48 hours.
Important: This is an original study guide, not a reproduction of a textbook or official examination paper.
Class 11 Mathematics Coursebook • Page 1
,1. Sets
Core ideas
A set is a well-defined collection of objects. Sets are commonly written using braces, for example A = {1,2,3}.
The empty set is ∅. A set with n elements has 2n subsets.
If every element of A belongs to B, then A ⊆ B. If A ⊆ B and A ≠ B, then A is a proper subset of B.
For sets A and B: union A∪B contains elements in either set; intersection A∩B contains elements common to both;
difference A−B contains elements in A but not B; complement A′ contains elements in the universal set U but not A.
De Morgan's laws: (A∪B)′ = A′∩B′ and (A∩B)′ = A′∪B′.
For finite sets, n(A∪B)=n(A)+n(B)−n(A∩B). For three sets, subtract the three pairwise intersections and add the triple
intersection.
Worked examples
Example: If A={1,2,3,4} and B={3,4,5}, find A∪B and A∩B.
Solution: A∪B={1,2,3,4,5}; A∩B={3,4}.
Example: A class has 35 students. 20 study Mathematics club activities, 18 study Science club activities, and 8 study both.
How many study at least one?
Solution: n(M∪S)=20+18−8=30.
Example: How many subsets does a 6-element set have?
Solution: 26=64 subsets.
Practice
1. Let A={2,4,6,8} and B={1,2,3,4}. Find A−B and B−A.
2. If n(A)=28, n(B)=19 and n(A∩B)=7, find n(A∪B).
3. How many proper subsets does a 5-element set have?
4. Verify one of De Morgan's laws using A={1,2,3}, B={3,4,5} in U={1,2,3,4,5,6}.
5. Give an example of two disjoint non-empty sets.
Class 11 Mathematics Coursebook • Page 2
, 2. Relations and Functions
Core ideas
The Cartesian product A×B is the set of ordered pairs (a,b) where a∈A and b∈B. If |A|=m and |B|=n, then |A×B|=mn.
A relation from A to B is any subset of A×B. A function f:A→B assigns exactly one element of B to each element of A.
The domain is the set of inputs; the codomain is the target set; the range is the set of actual outputs.
One-one means distinct inputs have distinct outputs. Onto means every codomain element is hit. Bijective means both.
Composition is (g■f)(x)=g(f(x)). An inverse function exists when a function is bijective, with f−1(f(x))=x.
Worked examples
Example: If f(x)=2x+3, find f(4).
Solution: f(4)=8+3=11.
Example: If f(x)=x+2 and g(x)=3x, find (g■f)(x).
Solution: g(f(x))=3(x+2)=3x+6.
Example: Find the inverse of f(x)=2x−5.
Solution: Let y=2x−5. Then x=(y+5)/2, so f−1(x)=(x+5)/2.
Practice
1. Find the domain of f(x)=1/(x−4).
2. If f(x)=x²+1, find f(−3).
3. Find (f■g)(x) for f(x)=x² and g(x)=x+1.
4. State whether f:{1,2,3}→{a,b,c}, f(1)=a,f(2)=b,f(3)=b is one-one and/or onto.
5. Find the inverse of f(x)=3x+7.
Class 11 Mathematics Coursebook • Page 3
Complete Coursebook • Concepts • Formulae • Solved Examples • Practice
ISC-oriented study guide
A structured course for building concepts, solving standard problems, and preparing for examinations.
How to use this book
Study the concept notes first, then cover the worked examples without looking at the solution. Attempt the practice set
under time pressure. Keep a formula sheet and an error log. Revisit questions you could not solve after 24–48 hours.
Important: This is an original study guide, not a reproduction of a textbook or official examination paper.
Class 11 Mathematics Coursebook • Page 1
,1. Sets
Core ideas
A set is a well-defined collection of objects. Sets are commonly written using braces, for example A = {1,2,3}.
The empty set is ∅. A set with n elements has 2n subsets.
If every element of A belongs to B, then A ⊆ B. If A ⊆ B and A ≠ B, then A is a proper subset of B.
For sets A and B: union A∪B contains elements in either set; intersection A∩B contains elements common to both;
difference A−B contains elements in A but not B; complement A′ contains elements in the universal set U but not A.
De Morgan's laws: (A∪B)′ = A′∩B′ and (A∩B)′ = A′∪B′.
For finite sets, n(A∪B)=n(A)+n(B)−n(A∩B). For three sets, subtract the three pairwise intersections and add the triple
intersection.
Worked examples
Example: If A={1,2,3,4} and B={3,4,5}, find A∪B and A∩B.
Solution: A∪B={1,2,3,4,5}; A∩B={3,4}.
Example: A class has 35 students. 20 study Mathematics club activities, 18 study Science club activities, and 8 study both.
How many study at least one?
Solution: n(M∪S)=20+18−8=30.
Example: How many subsets does a 6-element set have?
Solution: 26=64 subsets.
Practice
1. Let A={2,4,6,8} and B={1,2,3,4}. Find A−B and B−A.
2. If n(A)=28, n(B)=19 and n(A∩B)=7, find n(A∪B).
3. How many proper subsets does a 5-element set have?
4. Verify one of De Morgan's laws using A={1,2,3}, B={3,4,5} in U={1,2,3,4,5,6}.
5. Give an example of two disjoint non-empty sets.
Class 11 Mathematics Coursebook • Page 2
, 2. Relations and Functions
Core ideas
The Cartesian product A×B is the set of ordered pairs (a,b) where a∈A and b∈B. If |A|=m and |B|=n, then |A×B|=mn.
A relation from A to B is any subset of A×B. A function f:A→B assigns exactly one element of B to each element of A.
The domain is the set of inputs; the codomain is the target set; the range is the set of actual outputs.
One-one means distinct inputs have distinct outputs. Onto means every codomain element is hit. Bijective means both.
Composition is (g■f)(x)=g(f(x)). An inverse function exists when a function is bijective, with f−1(f(x))=x.
Worked examples
Example: If f(x)=2x+3, find f(4).
Solution: f(4)=8+3=11.
Example: If f(x)=x+2 and g(x)=3x, find (g■f)(x).
Solution: g(f(x))=3(x+2)=3x+6.
Example: Find the inverse of f(x)=2x−5.
Solution: Let y=2x−5. Then x=(y+5)/2, so f−1(x)=(x+5)/2.
Practice
1. Find the domain of f(x)=1/(x−4).
2. If f(x)=x²+1, find f(−3).
3. Find (f■g)(x) for f(x)=x² and g(x)=x+1.
4. State whether f:{1,2,3}→{a,b,c}, f(1)=a,f(2)=b,f(3)=b is one-one and/or onto.
5. Find the inverse of f(x)=3x+7.
Class 11 Mathematics Coursebook • Page 3
