Sets, functions and groups: Class 11 Mathematics Notes, Short Questions and Numericals

Sets and their operations, Venn diagrams, relations, functions and their types, inverse and composite functions, binary operations, and the idea of a group with simple examples and properties.

Key points

A set is a well defined collection of objects, called its elements or members. We write a ∈ A when a is in A and a ∉ A when it is not. A set can be written in tabular form, A = {1, 2, 3}, or in set builder form, A = {x | x is a natural number less than 4}. The set with no elements is the empty set ∅. A set with one element is a singleton.

A is a subset of B, written A ⊆ B, if every element of A is in B. If A ⊆ B and A ≠ B, then A is a proper subset. A set with n elements has 2ⁿ subsets, and 2ⁿ − 1 of them are proper. The set of all subsets of A is the power set P(A). The universal set U contains all the elements under discussion. Two sets are equal when they have the same elements, and they are equivalent when they have the same number of elements.

Operations on sets:

Union A ∪ B: the elements in A or B or both.
Intersection A ∩ B: the elements common to A and B. If A ∩ B = ∅ the sets are disjoint.
Difference A − B: the elements in A that are not in B.
Complement A′: the elements of U that are not in A, so A′ = U − A.

The laws of sets are like the laws of numbers. Commutative: A ∪ B = B ∪ A and A ∩ B = B ∩ A. Associative: (A ∪ B) ∪ C = A ∪ (B ∪ C). Distributive: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) and A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C). De Morgan's laws: (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′.

A Venn diagram shows the universal set as a rectangle and each set as a circle inside it. Shading the overlap of two circles shows A ∩ B, and shading both circles shows A ∪ B. For finite sets, n(A ∪ B) = n(A) + n(B) − n(A ∩ B). This rule is used to solve problems on groups of people with common interests.

The ordered pair (a, b) is a pair in which the order matters, so (a, b) ≠ (b, a) unless a = b. The Cartesian product A × B is the set of all ordered pairs (a, b) with a in A and b in B. If A has m elements and B has n elements, then A × B has mn elements. A relation from A to B is any subset of A × B. Its domain is the set of first entries and its range is the set of second entries.

A function f from A to B is a relation in which every element of A is related to exactly one element of B. We write f: A → B and y = f(x). A is the domain, B is the codomain, and the set of actual outputs is the range. A graph represents a function if no vertical line cuts it at more than one point. For f(x) = √(x − 3) the domain is x ≥ 3, because the square root of a negative number is not real.

Types of functions:

Into function: the range is a proper subset of the codomain.
Onto function: the range equals the codomain.
One to one (injective): different inputs give different outputs.
Bijective: both one to one and onto. Only a bijective function has an inverse.

For f: A → B and g: B → C, the composite function g∘f is defined by (g∘f)(x) = g(f(x)). In general g∘f ≠ f∘g. The inverse f⁻¹ undoes f, so f⁻¹(f(x)) = x. To find it, write y = f(x), solve for x in terms of y, and then swap the letters. The graph of f⁻¹ is the reflection of the graph of f in the line y = x.

A binary operation * on a set S gives, for any two elements a and b of S, another element a * b of S. This property is called closure. The operation is commutative if a * b = b * a and associative if (a * b) * c = a * (b * c). An element e is the identity if a * e = e * a = a for all a. The inverse of a is the element a′ with a * a′ = e.

A group (G, *) is a set G with a binary operation that satisfies four conditions: closure, associativity, an identity element in G, and an inverse in G for every element. If the operation is also commutative, the group is abelian. Examples are the integers under addition (identity 0, inverse of a is −a) and the non zero rational numbers under multiplication (identity 1, inverse of a is 1/a). The natural numbers under addition do not form a group, since there is no identity or inverse in the set. In a group the identity is unique, and each element has only one inverse.

Study tip: Draw a Venn diagram for every word problem on sets and fill in the overlap first, then the rest. For groups, write the four conditions in a list and tick them off one by one with a short check for each. Always state the identity and the inverse clearly, because most marks go to those two checks.

Definitions and flashcards

Say the meaning first, then open the card to check.

Set

A well defined collection of objects.

Power set

The set of all subsets of a set. It has 2ⁿ elements.

Cartesian product

The set A × B of all ordered pairs (a, b) with a in A and b in B.

Domain

The set of all inputs of a function or relation.

Bijective function

A function that is both one to one and onto.

Closure

The result of the operation on any two elements of the set is again in the set.

Identity element

An element e with a * e = e * a = a for every a.

Abelian group

A group in which the operation is commutative.

Formula sheet

Name	Formula	What it means
Union count	n(A ∪ B) = n(A) + n(B) − n(A ∩ B)	Used for two finite sets.
Number of subsets	2ⁿ	A set with n elements.
Size of a product	n(A × B) = n(A) × n(B)	Number of ordered pairs.
De Morgan's laws	(A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′	Complement of a union or an intersection.
Composite function	(g∘f)(x) = g(f(x))	Apply f first, then g.

Short questions with model answers

These are the kind of questions asked for 2, 3 and 5 marks. Try each one before opening the answer.

How many subsets and proper subsets does a set of 4 elements have? (2 marks)

The number of subsets is 2⁴ = 16. The number of proper subsets is 16 − 1 = 15, because the set itself is not a proper subset.

State De Morgan's laws. (2 marks)

For any sets A and B, (A ∪ B)′ = A′ ∩ B′. Also (A ∩ B)′ = A′ ∪ B′. The complement of a union is the intersection of the complements, and the complement of an intersection is the union of the complements.

Define a function and say when it has an inverse. (3 marks)

A function f from A to B is a rule that assigns to each element of A exactly one element of B. It has an inverse only when it is bijective. One to one means no two inputs share an output, and onto means every element of B is an output. Then each y in B comes from exactly one x in A, so the rule can be reversed.

Let A = {1, 2} and B = {a, b, c}. Write A × B and say how many relations from A to B exist. (3 marks)

A × B = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)}. It has 2 × 3 = 6 elements. A relation is any subset of A × B, so the number of relations is 2⁶ = 64.

List the four conditions for (G, *) to be a group. Show that the integers under addition satisfy them. (5 marks)

The conditions are closure, associativity, an identity and inverses. For the integers: the sum of two integers is an integer, so closure holds. (a + b) + c = a + (b + c), so addition is associative. The identity is 0, since a + 0 = a. The inverse of a is −a, which is an integer, since a + (−a) = 0. So the integers form a group under addition, and it is abelian because a + b = b + a.

Explain the difference between a one to one function and an onto function with an example. (5 marks)

A function is one to one if f(x₁) = f(x₂) forces x₁ = x₂. It is onto if every element of the codomain is the image of some element of the domain. Take f: R → R with f(x) = x². It is not one to one, since f(2) = f(−2) = 4. It is not onto, since negative numbers are never outputs. In contrast, f(x) = 2x + 1 from R to R is both one to one and onto.

Numericals with worked steps

Let A = {1, 2, 3, 4, 5, 6} and B = {4, 5, 6, 7, 8}. Find A ∪ B, A ∩ B and A − B, and check the count formula.

A ∪ B contains every element of either set: {1, 2, 3, 4, 5, 6, 7, 8}.
A ∩ B contains the common elements: {4, 5, 6}.
A − B contains the elements of A not in B: {1, 2, 3}.
Check: n(A) + n(B) − n(A ∩ B) = 6 + 5 − 3 = 8, and n(A ∪ B) = 8.

Answer: A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8}, A ∩ B = {4, 5, 6}, A − B = {1, 2, 3}, and the formula checks.

In a class of 40 students, 25 like cricket, 18 like hockey and 5 like neither. How many like both, and how many like cricket only?

Students who like at least one game = 40 − 5 = 35, so n(C ∪ H) = 35.
Use n(C ∪ H) = n(C) + n(H) − n(C ∩ H): 35 = 25 + 18 − x.
So x = 43 − 35 = 8.
Cricket only = 25 − 8 = 17.

Answer: 8 students like both games, and 17 like cricket only.

Find the inverse of f(x) = 2x + 3 and check it by finding f(f⁻¹(7)).

Write y = 2x + 3.
Solve for x: 2x = y − 3, so x = (y − 3)/2.
Swap the letters: f⁻¹(x) = (x − 3)/2.
f⁻¹(7) = (7 − 3)/2 = 2.
f(2) = 2(2) + 3 = 7, which is the starting value.

Answer: f⁻¹(x) = (x − 3)/2, and f(f⁻¹(7)) = 7.

On the integers define a * b = a + b − 1. Find the identity element and the inverse of 5.

Let e be the identity: a * e = a + e − 1 = a, so e = 1.
Check on the other side: e * a = 1 + a − 1 = a.
Let 5′ be the inverse of 5: 5 * 5′ = 5 + 5′ − 1 = 1.
So 5′ = 1 + 1 − 5 = −3.
Check: 5 * (−3) = 5 − 3 − 1 = 1.

Answer: The identity is 1 and the inverse of 5 is −3.

Common mistakes

Writing that a set with n elements has n² subsets. The correct number is 2ⁿ.
Mixing up ∈ and ⊆. An element belongs to a set (∈), while a set is contained in a set (⊆). So 2 ∈ {1, 2} but {2} ⊆ {1, 2}.
Forgetting that (a, b) and (b, a) are different ordered pairs, and so A × B is not the same as B × A in general.
Rejecting a rule as a function because an output repeats. Repeated outputs are allowed. The rule that breaks the definition is one input with two outputs.
Saying g∘f means apply g first. In g∘f, the function f is applied first and then g.

Practise this chapter

Practice questions and timed test Printable worksheet