Page 74 - Computer science 868 Class 12
P. 74
0 1 1 0 1
1 0 0 1 0
1 0 1 0 1
1 1 0 0 1
1 1 1 1 1
Referring C as C the Boolean expression of Sum term = A'.B'.C+A'.B.C'+A.B'.C'+A.B.C which is equivalent to three
in
variable XOR gate. The proof is given below:
A'.B'.C+A'.B.C'+A.B'.C'+A.B.C
= A'.(B'.C+B.C')+A.(B'.C'+B.C) [Distributive Law]
= A'.(B⊕C) + A.(B⊕C)'
= A⊕ B⊕C
Boolean expression for carry term = A'.B.C+A.B'.C+A.B.C'+A.B.C can be further minimised to A.B+B.C+C.A as done
below:
= A'.B.C+A.B'.C+A.B.C'+A.B.C+A.B.C+A.B.C [Idempotent Law]
= A'.B.C+A.B.C+A.B'.C+A.B.C+A.B.C'+A.B.C [Associative Law]
= B.C.(A'+A)+A.C.(B'+B)+A.B.(C+C') [Distributive Law]
= B.C+A.C+A.B [Complement Law]
= A.B+B.C+C.A
The logic circuit diagram of a full adder circuit is:
A Sum = A⊕B⊕C
B in
C in
A.B
A.B+B.C +C .A
in
in
B.C in
C .A
in
The full adder circuit can also be represented as two half adders connected by OR gate as follows:
Sum of full adder = A'.B'.C+A'.B.C'+A.B'.C'+A.B.C
= A'.(B'.C+B.C')+A.(B'.C'+B.C) [Distributive Law]
= A'.(B⊕C)+A.(B⊕C)'
= A⊕B⊕C
Carry of full adder = A'.B.C+A.B'.C+A.B.C'+A.B.C
= C.(A'.B+A.B')+A.B.(C'+C) [Distributive Law]
= C.(A⊕B)+A.B [Complement Law]
7272 Touchpad Computer Science-XII

