FOUNDATION MATHEMATICS NOTES
Q. Define sets.
Sets are a collection of well-defined objects or elements. The elements of a set can be
anything: numbers, letters, objects, or even other sets. Sets are usually denoted by capital
letters, and the elements inside a set are listed within curly braces { }.
Example: A= {1,2,3,} where 1,2 and 3 are the elements of set A.
Q. What is union and intersection of sets?
Union of Sets (𝐴 ∪ 𝐵)
The union of two sets A and B is the set of all elements that are in A, or in B, or in both. The
union includes all distinct elements from both sets.a
Example:
𝐴 = {1,2,3}, 𝐵 = {3,4} then (𝐴 ∪ 𝐵) = {1,2,3,4}
Intersection of Sets:
The intersection of two sets A and B is the set of elements that are common to both A and B.
Example:
𝐴 = {1,2,3}, 𝐵 = {2,3,4} then 𝐴 ∩ 𝐵 = {2,3}.
Q. Describe ways to describe a set.
A set can be described in the following ways:
1) Tabular Form: The elements of the set are listed, with each element separated by commas
and enclosed in curly braces { }.
Example: A={1,2,3,4}
2) Set-builder Notation: This method describes the set by stating the properties that its
members must satisfy.
Example: B={x | x is an even number}
3. Descriptive Form:
In this method, the set is described in words, specifying the nature or characteristics of its
elements.
Example:
"The set of all natural numbers less than 10."
, Q. Write a note on Venn Diagrams.
Venn diagrams are the diagrams that are used to represent the sets, relation between the sets
and operation performed on them, in a pictorial way. Venn diagram, introduced by John
Venn (1834-1883), uses circles (overlapping, intersecting and non-intersecting), to denote the
relationship between sets. A Venn diagram is also called a set diagram or a logic diagram
showing different set operations such as the intersection of sets, union of sets and difference
of sets. It is also used to depict subsets of a set.
Example:
𝐴∩𝐵 𝐴∪𝐵
Q. What are power sets ?
The power set is a set which includes all the subsets including the empty set and the original
set itself. It is usually denoted by P. Power set is a type of sets, whose cardinality depends on
the number of subsets formed for a given set. If set A = {x, y, z} is a set, then all its subsets
{x}, {y}, {z}, {x, y}, {y, z}, {x, z}, {x, y, z} and {} are the elements of power set, such as:
Power set of A, P(A) = {{x}, {y}, {z}, {x, y}, {y, z}, {x, z}, {x, y, z}}, where P(A) denotes
the power set.
Q. Write a note on base 10 binary base 5 octal number system.
1. Base 10 (Decimal System)
The base 10 number system, also known as the **decimal system*, is the most commonly
used number system in everyday life. It is called base 10 because it is based on ten digits: 0,
1, 2, 3, 4, 5, 6, 7, 8, and 9. It is the standard system for denoting integer and non-integer
numbers and is universally used in daily arithmetic and in most scientific and engineering
applications.
Q. Define sets.
Sets are a collection of well-defined objects or elements. The elements of a set can be
anything: numbers, letters, objects, or even other sets. Sets are usually denoted by capital
letters, and the elements inside a set are listed within curly braces { }.
Example: A= {1,2,3,} where 1,2 and 3 are the elements of set A.
Q. What is union and intersection of sets?
Union of Sets (𝐴 ∪ 𝐵)
The union of two sets A and B is the set of all elements that are in A, or in B, or in both. The
union includes all distinct elements from both sets.a
Example:
𝐴 = {1,2,3}, 𝐵 = {3,4} then (𝐴 ∪ 𝐵) = {1,2,3,4}
Intersection of Sets:
The intersection of two sets A and B is the set of elements that are common to both A and B.
Example:
𝐴 = {1,2,3}, 𝐵 = {2,3,4} then 𝐴 ∩ 𝐵 = {2,3}.
Q. Describe ways to describe a set.
A set can be described in the following ways:
1) Tabular Form: The elements of the set are listed, with each element separated by commas
and enclosed in curly braces { }.
Example: A={1,2,3,4}
2) Set-builder Notation: This method describes the set by stating the properties that its
members must satisfy.
Example: B={x | x is an even number}
3. Descriptive Form:
In this method, the set is described in words, specifying the nature or characteristics of its
elements.
Example:
"The set of all natural numbers less than 10."
, Q. Write a note on Venn Diagrams.
Venn diagrams are the diagrams that are used to represent the sets, relation between the sets
and operation performed on them, in a pictorial way. Venn diagram, introduced by John
Venn (1834-1883), uses circles (overlapping, intersecting and non-intersecting), to denote the
relationship between sets. A Venn diagram is also called a set diagram or a logic diagram
showing different set operations such as the intersection of sets, union of sets and difference
of sets. It is also used to depict subsets of a set.
Example:
𝐴∩𝐵 𝐴∪𝐵
Q. What are power sets ?
The power set is a set which includes all the subsets including the empty set and the original
set itself. It is usually denoted by P. Power set is a type of sets, whose cardinality depends on
the number of subsets formed for a given set. If set A = {x, y, z} is a set, then all its subsets
{x}, {y}, {z}, {x, y}, {y, z}, {x, z}, {x, y, z} and {} are the elements of power set, such as:
Power set of A, P(A) = {{x}, {y}, {z}, {x, y}, {y, z}, {x, z}, {x, y, z}}, where P(A) denotes
the power set.
Q. Write a note on base 10 binary base 5 octal number system.
1. Base 10 (Decimal System)
The base 10 number system, also known as the **decimal system*, is the most commonly
used number system in everyday life. It is called base 10 because it is based on ten digits: 0,
1, 2, 3, 4, 5, 6, 7, 8, and 9. It is the standard system for denoting integer and non-integer
numbers and is universally used in daily arithmetic and in most scientific and engineering
applications.
