1) The document discusses a lecture on logic and logical operations. It introduces logical variables, conditional statements, negation, contraposition, biconditionals, and logical operators like AND, OR, NOT. 2) Examples of using logic to represent situations are provided, like whether a server will crash given conditions like overheating, air conditioning being off, or cooler failing. 3) Boolean algebra rules and logic gate operations are defined. Truth tables are introduced as a way to represent logical statements and logical equivalence is discussed.

1. DR ATIF SHAHZAD
Fundamentals of
Computer Systems
IE-321
LECTURE #03

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Dr.AtifShahzad
RECAP
Digital System
What is a Numbering System?
Commonly Occurring Bases
Numbering System Examples
Base Conversion Procedure
Base Conversion
Number of Bits Required
Number of Elements represented
Q&A

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Dr.AtifShahzad
TODAY
Logic
Logical variables
Conditional, Negation, Contrapositive,Biconditional
AND, OR,NOT,XOR
Logic gates
TruthTables
Boolean Algebra
Examples
Q&A

4. Fundamentals of
Computer Systems
IE321

5. Logic
“ if the pool is warm, I’ll swim
What does it mean?
Yourinterpretations???
If the pool is cold, will I swim?
If I am swimming, pool is hot or cold?
If I am not swimming, does it mean that the pool is cold?
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6. Logic
“ if the pool is warm, I’ll swim
It is based on the validity of two
things, which can be mapped to
logical variables A and B :
A :The pool is warm.
B : I swim.
They’reeitherTrue or False
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A :
The pool is
warm.
B :
I swim.

7. Logic variables
“ if the pool is warm, I’ll swim
A :The pool is warm.
B : I swim.
A =True means a warm pool;
B = False means no swimming.
B can’t be half-true , because I can’t half swim.
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8. Conditional Operator
“ if the pool is warm, I’ll swim
Dependency between variables is expressed with →
, the conditional operator .
A → B is the idea that
A =True implies B =True :
So we have,
A → B : If the pool is warm, then I’ll
swim.
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9. Negation Operator
With more operators, different ideas can be expressed.
To negate ideas, we use ! , the negation
operator .
!A is the opposite of A
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! A : The pool is cold.
! B : I don’t swim.
A :
The pool is
warm.
B :
I swim.

10. Contrapositive Operator
GivenA → B and I didn’t swim,
what can be said
about the pool?
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11. Contrapositive Operator
GivenA → B and I didn’t swim,
what can be said about the pool?
A warm pool forces the swimming, so
without swimming, it’s impossible for
the pool to be warm
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12. Contrapositive Operator
GivenA → B and I didn’t swim,
what can be said about the pool?
A warm pool forces the swimming, so
without swimming, it’s impossible for
the pool to be warm
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13. Contrapositive Operator
Every conditional expression has a
contrapositive equivalent:
For any two variablesA and B,
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A → B is the same as
! B → ! A.

14. Contrapositive : Example
If you can’t write good code, you
haven’t read this article .
Its contrapositive is
if you read this article, you can write
good code .
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15. Biconditional
Be careful,
saying “if the pool is warm, I’ll swim”
doesn’t mean
I’ll only swim in warm water.
The statement promises nothing about
cold pools.
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In other words, A → B doesn’t mean B → A

16. Biconditional
To express both conditionals, use the biconditional :
A ↔ B : I’ll swim if and only if the
pool is warm.
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In other words, A → B doesn’t mean B → A

17. AND
A :You drank Cola.
B :You drank Juice.
A OR B :You drank.
A AND B :You drank mixing drinks.
A XOR B :You drank without mixing.
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18. OR,AND, XOR
A :You drank Cola.
B :You drank Juice.
A OR B :You drank.
A AND B :You drank mixing drinks.
A XOR B :You drank without mixing.
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19. Logical Operation:AND
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In a two variable case.
written asY=A.B
Dot (.) symbol is the common symbol of AND gate.
When both inputs are 1 then output is logically 1.
When both or any of the input is 0 then output is
logically 0.
We can also write AND Operation as:Y=AB

20. Logical Operation: OR
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In a two variable case.
written asY=A+B
Plus (+) symbol is the common symbol of OR gate.
When both inputs are 0 then output is logically 0.
When both inputs or any of one input is 1 then output
is logically 1.

21. Logical Operation: NOT
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It has only ONE input.
NOT Operation is also named as complementation.
It changes any input to it’s complement, like 1 to 0 &
0 to 1.
It is also written as A= Ā.
It is also called as INVERTER.

22. Logical Operation: XOR
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In a two variable case.
An XOR gate produces 0 if its both inputs are the
same, and 1 otherwise

23. Logical Operations with twoVariables
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24. Use of Logic Gates
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25. Use of Logic Gates
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Calculating 2+3=5
In binary 10+11=101

26. BooleanAlgebra: Some Rules
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Operations with 0 & 1
• X+0 = X
• X+1 = 1
Idempotent Laws
• X+X = X
• (X΄)΄ = X
Laws of Complementarity
• X+X΄ = 1
Operations with 0 & 1
• X.0 = 0
• X.1 = X
Idempotent Laws
• X.X = X
Laws of Complementarity
• X.X΄ = 0

27. BooleanAlgebra: Some Rules
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Commutative Laws
• X.Y =Y.X
Associative Laws
• (XY)Z = X(YZ) = XYZ
Distributive Laws
• X+YZ = (X+Y)(X+Z)
SimplificationTheorems
• (X+Y)(X+Y΄) = X
• X(X+Y) = X
• XY΄+y = X+Y

28. BooleanAlgebra: Some Rules
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Commutative Laws
• X+Y =Y+X
Associative Laws
• (X+Y)+Z = X+(Y+Z)
Distributive Law
• X(Y+Z) = XY+XZ
SimplificationTheorems
• XY + XY΄ = X
• X+XY = X
• (X+Y΄)Y = XY

29. BooleanAlgebra: DeMorgan
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DEMORGAN’S Law
• (X+Y+Z+…)΄ = X΄Y΄Z΄
DEMORGAN’S Law
• (XYZ…)΄ = X΄+Y΄ +Z΄+…

30. Example
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A server crashes if it’s overheating while
the air conditioning is off.
It also crashes if it’s overheating and its
chassis cooler fails.
In which conditions does the server work?

31. Example (contd.)
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A : Server overheats.
B :Air conditioning off.
C : Chassis cooler fails.
D : Server crashes.

32. Example (contd.)
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(A AND B) OR (A AND C)→ D
A : Server
overheats.
B : Air
conditioning off.
C : Chassis
cooler fails.
D : Server
crashes.

33. Example (contd.)
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(A AND B) OR (A AND C)→ D
ByAssociativity,
A AND (B OR C) → D
A : Server
overheats.
B : Air
conditioning off.
C : Chassis
cooler fails.
D : Server
crashes.

34. Example (contd.)
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(A AND B) OR (A AND C)→ D
ByAssociativity,
A AND (B OR C) → D
A : Server
overheats.
B : Air
conditioning off.
C : Chassis
cooler fails.
D : Server
crashes.
The server works when ( ! D ). The contrapositive reads:
(! D → ! (A AND (B OR C))

35. Example (contd.)
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A : Server
overheats.
B : Air
conditioning off.
C : Chassis
cooler fails.
D : Server
crashes.
(! D → ! (A AND (B OR C))
By Demorgan’s law,
! D → !A OR ! (B OR C)
Again applying DeMorgan’s law
! D → !A OR ( ! B AND ! C)

36. Example (contd.)
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A : Server
overheats.
B : Air
conditioning off.
C : Chassis
cooler fails.
D : Server
crashes.
! D → !A OR ( ! B AND ! C)
This expression tells us that
whenever the server works,
either ! A
(it’s not overheating),
or ! BAND ! C
(both air conditioning and chassis cooler
are working).

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NEVER hesitate to
contact should you
have any question
Dr.Atif Shahzad