Basic Concept of discrete math. I discuss a few topic of basic discrete math. I think you get a clear concept. Topics are definition,story of discrete math, Importance of discrete math, and many other basic topic.

1. Final Year Defense
Contents-
•What is Discrete mathematics.
•Story of the Discrete mathematics.
•Importance of Discrete mathematics for computer
science students.
•Topics of Discrete mathematics

2. Final Year Defense
What is discrete mathematics or what’s the mean
of Discrete mathematics?
• Means Different or Disconnect..
• Is not the name of branches.
• Description of set of branch.
• Common feature
• Continuous.
Class Presentation

3. Final Year Defense
Story of Discrete math
• In 26 Mar 1913.
• Paul Erdos.
• Hangarian.
• Found the field of the discrete.
• Foundation of computer science.
• One of the most prolific.
3
Class presentation

4. Final Year Defense
Why Discrete mathematics is important for
Computer Science students?
• Represent and manipulate of data.
• Is not programming.
• Is not software engineering.
• is about problem solving.
• Solid background.
Class presentation4

5. Final Year Defense
Topics
• Propositional Logic.
• Propositional Equivalence.
• Set.
• Relation.
• Graph.
Class presentation5

6. Topic
Definition of Propositional Logic
AND Operation
OR Operation
XOR Operation

7. Propositional Logic: Propositional logic is a statement that is truth
value “True” or a truth value “False”.
Example: P: Today is sun day. (Truth Value “True”)
¬P: Today is not sun day. (Truth Value “False”)
Truth Table:
P ¬P
T F
F T

8. AND Operation (Л): True if both statement are true Truth Table:
Example: P: Today is Tuesday.
Q: It is presentation hour.
PЛQ: Today is Tuesday and it is presentation hour.
OR Operation (V): True if any statement is true
Example: P: Today is Tuesday. Truth Table:
Q: It is raining.
PVQ: Today is Tuesday or it is raining.
P Q PЛQ
T T T
T F F
F T F
F F F
P Q PVQ
T T T
T F T
F T T
F F F
Example
Example

9. XOR Operation (⊕):True if different statement is true.
Example: P : You will get laptop.
Q : Equal amount of money.
P⊕Q : You will get laptop or equal amount of money.
Truth Table:
P Q P⊕Q
T T F
T F T
F T T
F F F

10. Topic
Truth table of some Logical Equivalence
1. Commutative Law
2. Double Negation Law
3. Absorption Law

11. P Q P˅Q Q˅P P˄Q Q˄P
T T T T T T
T F T T F F
F T T T F F
F F F F F F
*COMMUTATIVE LAWS:
P˅Q≡Q˅P
P˄Q≡Q˄P
Truth Table:
FROM THE ABOVE TRUTH TABLE WE CAN PROVE THAT P˅Q≡Q˅P AND P˄Q≡Q˄P

12. P ¬P ¬ (¬P)
T F T
F T F
*Double Negation Law:
¬ (¬P)≡P
Truth Table:
From the above truth table we can prove that ¬ (¬P)≡P

13. P Q P˄Q P˅Q P˅(P˄Q) P˄(P˅Q)
T T T T T T
T F F T T T
F T F T F T
F F F F F F
*Absorption Laws:
P ˅ (P˄Q) ≡ P
P ˄ (P˅Q) ≡ P
Truth Table:
From the above truth table we can prove that P˅(P˄Q) ≡P and P˄(P˅Q) ≡P

14. Set
A group of objects, numbers, thoughts,
etc.
are called sets. Each object in the set is
called element or member of the set.
For example
N = {1,2,3,4,5,6,7,8,9} is set here. The
elements
of the set are 1,2,3,4,5,6,7,8,9.

15. Union Set
Combining all the elements of any two sets is
called the Union of those sets.
For example
A = { 1, 2, 4, 6} and B = { 4, a, b, c, d, f}
A ∪ B = { 1, 2, 4, 6, a, b, c, d, f}
Notice that it is perfectly ok to write 4 once or twice
We write it once. Because this element is same.
A U B

16. Intersection Set
Intercept of two set A and B is the set of
members who are members of both set A and B.
For example
To make it easy, notice that what they have in
common is in bold
A = {b, 1, 2, 4, 6} and B = { 4, a, b, c, d, f }
A ∩ B = {4, b}

17. Power Set
Power set of a set is a set of all subset of the set.
For example
For the set S= {1,2,3} this means:
subsets with 0 elements: {Φ} (the empty set)
subsets with 1 element: {1}, {2}, {3}
subsets with 2 elements: {1,2}, {1,3}, {2,3}
subsets with 3 elements: {1,2,3}
Hence: P(S) = {Φ,{1}, {2}, {3},{1,2}, {1,3},
{2,3}, {1,2,3}}

18. SUB SET
What is SUB
SET?

19. If every member of set A is also member of set B, then A
is a subset of B . We write A ⊆ B.
Example: A={1,3,5,7,9},
B={1,2,3,4,5,6,7,8,9}.
A ⊆ B
Because, there all element are same.
Class Presentation

20. A ⊆ B
A ⊆ B⊆C
C
B
B⊆C

21. NOW, I AM DISCUSS
ABOUT PROPER SUBSETS
If A is subset of B(A⊆B).But A is not equal to B , then we
say A is proper subset of B. A⊂B.
Exampule : A= {a,b,c,d,e},
B={a,b,c,d,e,f,g,h,I,j,k}.
Here, A⊂B=T.
Because , there same as subject but not A=B.
So we can say B⊂A=F.

22. Reflexive
Reflexive is a tarm of a relation. A relation R on a
set A is called reflexive if(a, a) ∈ R for every
element a ∈ A.
Rules:
A=B (A & B are same like as 1,1)

23. Example :
Consider the following A= {1,2,3,}
R={(1,1),(1,2),(1,3),(2,2),(2,3),(3,3)}
The relation R is reflexive because all contain pairs
of the from (a=b) namely (1,1)(2,2)(3,3).

24. Symmetric
Symmetric is the another term of relation.
A relation R on a set A is called symmetric . If (a,b) ∈ R
whenever (b,a) ∈ R , for all (a,b) ∈ A.
Rule: (a,b) ∈ R then must be,
(b,a) ∈ R.

25. Example:
R = {(1,1),(1,2),(2,1)}
The relation R is a symmetric because in this case
(a,b) belongs to the relation and must (b,a) belongs
to that relation.

26. GRAPH
1. DEFINITION OF GRAPH
2. CONDITION TO BE A GRAPH
3. DIFFERENT TYPES OF GRAPH
4. HANDSHAKING THEORY

27. .

28. Different Types of Graph
Undirected Graph: If there is not any direction in a graph then it is called undirected graph.
Example:
Directed Graph: If there is direction in a graph then it is called directed graph.
Example: A
C
B
E
D
A
C D
E
B

29. HAND SHAKING THEORY
THEORY: An undirected graph has an even number of odd degrees.
There are two kinds of degree. in-degree and out- degree.
ƩDEG-(V)=ƩDEG+(V)=|E| 6=6=6
A
C D
B
E
deg+(D)=1
deg-(D)=1
deg+(E)=1
deg-(E)=1
deg-(A)=1
deg+(A)=2
deg-(B)=2
deg+(B)=0
deg-(C)=1
deg+(C)=2