Download 10th Class Computer Science Notes (Unit # 06) for Federal Board (FBISE) Islamabad.
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THEORY
Question: What is Boolean algebra?
Boolean algebra is a mathematical logic. It
is presented by George Boole. All electronic switches, devices and circuits
must have two states. ON and OFF. Therefore this is also termed as switching
algebra. Boolean algebra uses algebraic notation to express logical
relationship.
Boolean
Constants:
i)
0 if not 1
ii)
1 if not 0
Boolean
Variables:
Variables are represented by letters of
alphabets having values equal to 0 and 1
Question: What do you understand by the
logical operations AND and OR?
AND
Operation:
In Boolean Algebra AND operation represents
logical multiplication. i.e.
Z = X AND Y = X.Y = XY
We can show it in truth table form as:
X
Y
Z
0
0
0
0
1
0
1
0
0
1
1
1
OR
Operation:
In Boolean Algebra AND operation represents
logical addition. i.e.
Z = X OR Y = X + Y
We can show it in truth table form as:
X
Y
Z
0
0
0
0
1
1
1
0
1
1
1
1
Question: What is a Truth Table?
Construct a Truth Table for AND and NOT of AND operations for the three
variables X, Y and Z.
A truth table is a table that shows the
result of a Boolean expression for all the possible combinations of the values
given to the variables used.
Truth
Table for AND Operation:
X
Y
Z =
A.B
0
0
0
0
1
1
1
0
1
1
1
1
Truth
Table for NOT Operation:
X
Z =
X
0
1
1
0
Question: State and Prove the following
Laws:
Idempotent
Law
a) A
+ A = A b) A . A = A
a)
L.H.S R.H.S
= A
+ A A
= (A
+ A) . 1
= (A
+ A) . (A + A)
= A
+ (A . A)
= A
+ 0
= A
b)
L.H.S R.H.S
= A
. A A
= A
. A + 0
= A
. A + A . A
= A
. (A + A)
= A
. 1
= A
Involution
Law
It states that double complementation has
cancellation effect. This can be proved by the method of perfect induction as
shown in truth table below:
A
A
A
0
1
0
1
0
1
Absorption
Law
a) A
+ (A.B) = A b) A . (A + B) = A
a)
L.H.S R.H.S
= A
+ (A . B) A
= (A
. 1) + (A . B)
= A
. (1 + B)
= A
. 1
= A
b)
L.H.S R.H.S
= A
. (A + B) A
= (A
+ 1) . (A + B)
= A
+ (0 . B)
= A
+ 1
= A
=*=*=*=*=
THEORY
Question: What is Boolean algebra?
Boolean algebra is a mathematical logic. It
is presented by George Boole. All electronic switches, devices and circuits
must have two states. ON and OFF. Therefore this is also termed as switching
algebra. Boolean algebra uses algebraic notation to express logical
relationship.
Boolean
Constants:
i)
0 if not 1
ii)
1 if not 0
Boolean
Variables:
Variables are represented by letters of
alphabets having values equal to 0 and 1
Question: What do you understand by the
logical operations AND and OR?
AND
Operation:
In Boolean Algebra AND operation represents
logical multiplication. i.e.
Z = X AND Y = X.Y = XY
We can show it in truth table form as:
X

Y

Z

0

0

0

0

1

0

1

0

0

1

1

1

OR
Operation:
In Boolean Algebra AND operation represents
logical addition. i.e.
Z = X OR Y = X + Y
We can show it in truth table form as:
X

Y

Z

0

0

0

0

1

1

1

0

1

1

1

1

Question: What is a Truth Table?
Construct a Truth Table for AND and NOT of AND operations for the three
variables X, Y and Z.
A truth table is a table that shows the
result of a Boolean expression for all the possible combinations of the values
given to the variables used.
Truth
Table for AND Operation:
X

Y

Z =
A.B

0

0

0

0

1

1

1

0

1

1

1

1

Truth
Table for NOT Operation:
X

Z =
X

0

1

1

0

Question: State and Prove the following
Laws:
Idempotent
Law
a) A
+ A = A b) A . A = A
a)
L.H.S R.H.S
= A
+ A A
= (A
+ A) . 1
= (A
+ A) . (A + A)
= A
+ (A . A)
= A
+ 0
= A
b)
L.H.S R.H.S
= A
. A A
= A
. A + 0
= A
. A + A . A
= A
. (A + A)
= A
. 1
= A
Involution
Law
It states that double complementation has
cancellation effect. This can be proved by the method of perfect induction as
shown in truth table below:
A

A

A

0

1

0

1

0

1

Absorption
Law
a) A
+ (A.B) = A b) A . (A + B) = A
a)
L.H.S R.H.S
= A
+ (A . B) A
= (A
. 1) + (A . B)
= A
. (1 + B)
= A
. 1
= A
b)
L.H.S R.H.S
= A
. (A + B) A
= (A
+ 1) . (A + B)
= A
+ (0 . B)
= A
+ 1
= A
=*=*=*=*=
COMMENTS