Discrete mathematics deals with discrete, separated objects rather than continuous ones. It examines countable and finite mathematical structures. Some key topics in discrete mathematics include set theory, logic, number theory, graph theory, and combinatorics. Set theory involves defining and examining sets and their relationships. A set is a collection of distinct elements. Sets can be defined by listing elements or using set-builder notation. Operations on sets include union, intersection, complement, difference, and Cartesian product. The power set of a set contains all its subsets. [END SUMMARY]

1. Unit 18
Discrete Mathematics

2. Why Mathematics?
Design efficient computer systems.
 How did Google manage to build a fast search engine?
 What is the foundation of internet security?
algorithms, data structures, database,
parallel computing, distributed systems,
cryptography, computer networks…
Logic, number theory, counting, set, graph theory…

3. 3
What Is Discrete Mathematics?
• What it isn’t: continuous
• Discrete: consisting of distinct or unconnected
elements
• Countably Infinite
• Definition Discrete Mathematics
• Discrete Mathematics is a collection of
mathematical topics that examine and use finite or
countably infinite mathematical objects.

4. What is Discrete Mathematics?
• Discrete mathematics is the branch of mathematics handling objects
that only considers distinct, separated values.
• It is steadily being applied in the multiple domains of mathematics
and computer science. It is accounted as a very effective approach for
developing and problem-solving strength.
• Discrete Mathematics focuses on the systematic study of
Mathematical structures that are essentially discrete in nature and
does not demand the belief of continuity.

5. What is Discrete Mathematics?
• In simple words, discrete mathematics gives an individual the ability
to understand mathematical language that can be learned through
various branches of it.

6. Set Theory
• A set is a collection of some items (elements). We often use capital
letters to denote a set. To define a set we can simply list all the
elements in curly brackets { } separated by commas.
• for example to define a set A that consists of the two elements ♣ and ♢,
we write A={♣,♢}
• To say that ♢ belongs to A, we write ♢∈A, where "∈" is pronounced
"belongs to." To say that an element does not belong to a set, we use ∉.
For example, we may write ♡∉A.
• Note that ordering does not matter, so the two
sets {♣,♢}{♣,♢} and {♢,♣}{♢,♣} are equal.
A set is a collection of things (elements).

7. Set Theory

8. Set Theory
Set Builder Notation
• We can also define a set by mathematically stating the properties satisfied by the
elements in the set. In particular, we may write
• A={x|x satisfies some property}A={x|x satisfies some property}
or
A={x:x satisfies some property}A={x:x satisfies some property}
The symbols "|" and ":" are pronounced "such that."
Example
•If the set C is defined as C={x|x∈Z,−2≤x<10}C={x|x∈Z,−2≤x<10}, then C={−2,−1,0,⋯,9}
• If the set D is defined as D={x2|x∈N}D={x2|x∈N}, then D={1,4,9,16,⋯}D={1,4,9,16,⋯}.
•The set of rational numbers can be defined as Q={a/b|a,b∈Z,b≠0}Q={ab|a,b∈Z,b≠0}.

9. Sep 2021 Unit 18- Discrete Mathematics 9
Set Theory
Examples for Sets
• “Standard” Sets:
• Natural numbers N = {0, 1, 2, 3, …}
• Integers Z = {…, -2, -1, 0, 1, 2, …}
• Positive Integers Z+ = {1, 2, 3, 4, …}
• Real Numbers R = {47.3, -12, , …}
• Rational Numbers Q = {1.5, 2.6, -3.8, 15, …}

10. Sep 2021 Unit 18- Discrete Mathematics 10
Set Theory
Examples for Sets
• A =  “empty set/null set”
• A = {z} Note: zA, but z  {z}
• A = {{b, c}, {c, x, d}}
• A = {{x, y}}
Note: {x, y} A, but {x, y}  {{x, y}}
• A = {x | P(x)}
“set of all x such that P(x)”
• A = {x | xN  x > 7} = {8, 9, 10, …}
“set builder notation”

11. Set Theory
• Venn Diagrams
Venn diagrams are very useful in
visualizing relation between sets. In
a Venn diagram any set is depicted by a
closed region.
• In this figure, the big rectangle shows the
universal set S. The shaded area shows
another set A

12. Set Theory
Subsets
•A  B “A is a subset of B”
•A  B if and only if every element of A is also an
element of B.
• Useful rules:
•   A for any set A
• A  A for any set A
• Equivalently, we say B is a superset of A, or B ⊇ A. .
•Examples:
Venn Diagram for two sets A and B, where B⊂A.

13. Set Theory
• Proper Subsets
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.
• A proper subset is denoted by ⊂
• If A and B are two sets, then A is called the proper
subset of B if A ⊆ B but B ⊇ A i.e., A ≠ B
example;
1) A = {1, 2, 3, 4}, Here n(A) = 4,
B = {1, 2, 3, 4, 5}, Here n(B) = 5
We observe that, all the elements of A are present in B but the
element ‘5’ of B is not present in A.
So, we say that A is a proper subset of B.
Symbolically, we write it as A ⊂ B
example;
• 2. A = {p, q, r}
B = {p, q, r, s, t}
Here A is a proper subset
of B as all the elements of
set A are in set B and also
A ≠ B.
Notes:
• No set is a proper subset of itself.
• Null set or ∅ is a proper subset of
every set.
Sep 2021 Unit 18- Discrete Mathematics 13

14. Set Theory
• How many subsets and
proper subsets does a set
have?
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.
Example
If set A has the
elements, A = {a, b},
then what is the subset
& proper subset of the
given set?

15. Sep 2021 Unit 18- Discrete Mathematics 15
Set Theory
• Power Set
• The power set is said to be the collection of all the subsets. It is represented by P(A).
• If A is set having elements {a, b}. Then the power set of A will be;
• P(A) = {∅, {a}, {b}, {a, b}}
• For example;
If A = {p, q} then all the subsets of A will be
P(A) = {∅, {p}, {q}, {p, q}}
Number of elements of P(A) = n[P(A)] = 4 =2^2
In general, n[P(A)] = 2^m where m is the number of elements in set A.

16. Sep 2021 Unit 18- Discrete Mathematics 16
Set Theory
• Universal Set
• A set which contains all the elements of other given sets is called
a universal set. The symbol for denoting a universal set is ∪ or ξ.
• For example;
If A = {1, 2, 3} B = {2, 3, 4} C = {3, 5, 7}
then U = {1, 2, 3, 4, 5, 7}
[Here A ⊆ U, B ⊆ U, C ⊆ U and U ⊇ A, U ⊇ B, U ⊇ C]

17. Set Theory
• Set Operations
• The union of two sets is a set containing all
elements that are in A or in B
• For example;
{1,2}∪{2,3}={1,2,3}
Thus, we can write x∈(A∪B) if and only
if (x∈A) or (x∈B)
Note that A∪B=B∪A
Sep 2021 Unit 18- Discrete Mathematics 17

18. Set Theory
• Set Operations
The intersection of two
sets A and B, denoted by A∩B,
consists of all elements that are both
in A and B
• For example;
{1,2}∩{2,3}={2}
Sep 2021 Unit 18- Discrete Mathematics 18

19. Set Theory
• Set Operations
The complement of a set A,
denoted by Ac or A¯ is the set of
all elements that are in the
universal set S but are not in A
Sep 2021 Unit 18- Discrete Mathematics 19

20. Set Theory
• Set Operations
The difference (subtraction) is
defined as follows. The
set A−B consists of elements that
are in A but not in B
For example
A={1,2,3} and B={3,5},
then A−B={1,2}.
Sep 2021 Unit 18- Discrete Mathematics 20
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21. Set Theory
• Set Operations
i.e., their intersection is the empty
set, A∩B=∅A∩B=∅.
Sep 2021 Unit 18- Discrete Mathematics 21
i
Two sets A and B are mutually
exclusive or disjoint if they do not have
any shared elements;

22. Set Theory
A Cartesian product
of two sets A and B, written as A×B, is the
set containing ordered pairs from A and B.
That is, if C=A×B, then each element
of C is of the form (x,y),
where x∈A and y∈B:
A×B={(x,y)|x∈A and y∈B}.
Sep 2021 Unit 18- Discrete Mathematics 22
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For example,
If A={1,2,3} and B={H,T},
then
A×B={(1,H),(1,T),(2,H),(2,T)
,(3,H),(3,T)}.
A×B is not the same as B×A.