This document provides an introduction to sets and subsets in mathematics. It defines what a set is as a well-defined collection of distinct objects or elements. It explains that a set is well-defined if its elements share a common property to determine if an object belongs to the set. The document also defines universal sets, subsets, proper subsets, the null or empty set, cardinality or number of elements in a set, and power sets, which are the collection of all subsets of a given set. It provides examples of subsets, proper subsets, and calculating the number of subsets for different sets.

1. HELLO!
WELCOME TO
GRADE 7
MATHEMATICS
1

2. INTRODUCTION
ON SETS

3. In each collection of object on each next
slides, find the one that is not part of the
group.
1. boat, kalesa, car, bus, airplane
The set contains the kinds of
transport that uses gas except for
kalesa.
2. carabao, chicken, cow, pig, goat The set contains with four legs except
for chicken.
3. hexagon, quadrilateral, rectangle, rhombus, square
The set contains with four sides except
for hexagon.

4. “
4
WHAT IS
SET?

5. 5
a well – defined
collection of distinct
objects, these
objects called
elements.
WELL – DEFINED- means that
the elements of a set share a
common property which helps
us to determine if an object
belongs to that set or not

6. 6
WELL - DEFINED SET OR NOT?
1. The set of all multiples of 5
Well-defined
2. The set of all large number
Not Well-defined

7. 7
7
WHAT IS
UNIVERSAL
SET?
A universal set (usually denoted by U) is a
set which has elements of all the related
sets, without any repetition of elements.
Say if A and B are two sets, such as A =
{1,2,3} and B = {1,a,b,c}, then the
universal set associated with these two
sets is given by U = {1,2,3,a,b,c}.

8. 8
WHAT IS
SUBSET?
Set A is a subset of set B, written as
B⊆A, if and only if every element in A
is also an element in B.
B={1,2,3,4,5,6,8,9}
A={1,3,6,8}
C={2,3,4,5}
D={5,8,9}

9. 9
What are Proper Subsets?

10. 10
Set A is considered to be a proper subset of Set B if Set B
contains at least one element that is not present in Set A.
Example: If set A has elements as {12, 24} and set B has
elements as {12, 24, 36}, then set A is the proper subset of B
because 36 is not present in the set A.
Proper Subset Symbol
A proper subset is denoted by ⊂ and is read as ‘is a proper
subset of’. Using this symbol, we can express a proper subset
for set A and set B as; A ⊂ B

11. Null set or empty
set
A set with no
element. In symbol,
it is written as { } or
∅.
The cardinality of a set A,
denoted by n(A), is the
number of elements in the
set. Thus, in A = {a, e, I, o u}
n(A) = 5 because set A
contains 5 elements.
11
Cardinality

12. How many subsets and proper subsets
does a set have?
12
If a set has “n” elements, then
the number of subset of the
given set is 2n and the number
of proper subsets of the given
subset is given by 2n-1.
How many subsets in the given
set A,
A = { m, a, t, h}?
P(A) = 24, P(A) = 16
Therefore, there are 16
subsets in set A, to check and
list all the subsets, let us find
out on the next slides
Power Set
The power set is said to be the
collection of all the subsets. It is
represented by P(A).
|P(A)| = 2n

13. 13
List of all subsets of set A
1. {m, a, t, h} 11. {t, h}
2. {m, a, t} 12. {m}
3. {m, a, h} 13. {a}
4. {m, t, h} 14. {t}
5. {a, t, h} 15. {h}
6. {m, a} 16. { }
7. {m, t}
8. {m, h} Those are the subsets of Set A, A = {m, a, t, h}
9. {a, t}
10.{a, h}
A = {m, a, t, h}

14. 14
LET US TRY
Determine all the subset of each set. Do it with
your self.
1. D = {tail, head}
2. O = {1, 2, 3}
We will check our solution on the next slide.

15. 15
Solution: Example 1
First, let us check the power set or the number of subsets
on the given set.
Given set: D = {tail, head}
N = 2
P(D) = 22, P(D) = 4 subsets
List of all subsets
1. {tail, head} there are 4 subsets in set D.
2. {tail}
3. {head}
4. { }

16. 16
First, let us check the power set or the number of subsets
on the given set.
Given set: O = {1,2, 3}
N = 3
P(O) = 23, P(0) = 8 subsets
List of all subsets
1. {1, 2, 3} 5. {1} there are 8 subsets on set O.
2. {1, 2} 6. {2}
3. {1, 3} 7. {3}
4. {2, 3} 8. { }
Solution: Example 2

17. 17