bs

1. (BC + AD) (AB + CD)
= BC.AB + BC.CD + AD.AB + AD.CD
= 0 + 0 + 0 + 0
= 0

2. ( Ā + C) + ( B + D̄ )
= A̿.C + B.D̿
= AC + BD

3. (BC + AD) (AB + CD)
= ABB̄C̄ + BCC̄D̄ + ĀAB̄D + ĀCDD̄
= A.0.C̄ + B.0.D̄ + 0.B̄.D + Ā.C.0
= 0 + 0 + 0 + 0
= 0

4. A+B.B+C
= (A + B) (B+C)
= AB + AC + BB + BC
= AB + AC + B + BC
= AB + AC + B(1+C)
= AB + AC + B
= B(A + 1) + AC
= B.1 + AC
= B + AC

5. AB̄C + ĀB̄C + ABC
= B̄C (A + Ā) + ABC
= B̄C + ABC
= B̄C(1 + A) + ABC
= B̄C + AB̄C + ABC
= B̄C + AC (B̄ + B)
= B̄C + AC

6. ABC + ĀBC̄ + ĀB̄C̄ + AB̄C
= ABC + AB̄C + ĀBC̄ + ĀB̄C̄
= AC (B + B̄) + ĀC̄ (B + B̄)
= AC + ĀC̄

7. প্রমাণ করো যে, (x + y) (x + z) (y + z) = (x + y) (x + z)

L.H.S. = (x+y) (x + z) (y + z)
= (xx + xz + xy + yz) (y + z)
= (0 + xz + xy + yz) (y + z)
= xyz + xzz + xyy + xyz + yyz + yzz
= xyz + xz + xy + xyz + yz
= xz (y + 1) + xy (1 + z) + yz
= xz + xy + yz

R.H.S. = (x+y) (x + z)
= xx + xz + xy + yz
= 0 + xz + xy + yz
= xz + xy + yz
L.H.S = R.H.S [Proved]

8. প্রমাণ করো যে,
(B + C̄) (B̄ + C) + (Ā + B + C̄ ) = BC + B̄ (C̄ + A)
L.H.S. = (B + C̄) (B̄ + C) + (Ā + B + C̄ )
= BB̄ + BC + B̄C̄ + CC̄ + AB̄C
= 0 + BC + B̄C̄ + 0 + AB̄C
= BC + B̄ (C̄ + AC)
= BC + B̄ (C̄ + AC)
= BC + B̄ (C̄(1 + A) + AC)
= BC + B̄ (C̄ + AC̄ + AC)
= BC + B̄ (C̄ + A (C̄ + C))
= BC + B̄ (C̄ + A)
= R.H.S.
L.H.S = R.H.S [Proved]

9. প্রমাণ করো যে, ((M + N̄) (M̄ + N) ) = M̄N + MN̄
L.H.S. = ((M + N̄) (M̄ + N) )
= (M + N̄) + (M̄ + N)
= M̄N + MN̄
= R.H.S.
L.H.S. = R.H.S. [Proved]

10. প্রমাণ করো যে, ĀBC̄ + ABC̄ + BC̄D = BC̄
L.H.S. = ĀBC̄ + ABC̄ + BC̄D
= BC̄ (Ā + A + D)
= BC̄ (1 + D)
= BC̄
= R.H.S.
L.H.S = R.H.S [Proved]

11. প্রমাণ করো যে, (A + B) (Ā + B̄) = 0
L.H.S. = (A + B) (Ā + B̄)
= ĀB̄A̿B̿
= ĀB̄AB
= 0
= R.H.S.
L.H.S = R.H.S [Proved]

12. (x̄y + xyz̄)
= x̄y . xyz̄
= (x + y) (x + x + z)
= xx + xy + xz + x y + y y + yz
= xy + x y + xz + y + yz
= y (x + x) + xz + y (1 + z)
= y+ y + xz
= y + xz

13. (x̄y + xyz̄).(ȳ + xz)
= x̄yȳ + x̄yxz + xyz̄ȳ + xyz̄xz
= 0 + 0 + 0 + 0
= 0

14. প্রমাণ করো যে, xy + xyz + y + xz = 1 + xz
= xy + xyz + y + xz (1 + y)
= xy + xyz + xyz + y + xz
= xy + xy (z + z) + y + xz
= xy + xy + y+ xz
= y (x + x) + y + xz
= y + y + xz
= 1 + xz
= R.H.S.
L.H.S = R.H.S [Proved]

15. প্রমাণ করো যে , A ⊕ B = AB + ĀB̄
L.H.S. =
= ĀB + AB̄
= ĀB . AB̄
= (A + B̄).(Ā + B)
= AĀ + AB + B̄Ā + B̄B
= 0 + AB + ĀB̄ + 0
= AB + ĀB̄
= R.H.S.
L.H.S = R.H.S [Proved]

16. প্রমাণ করো যে, (x + y) (x + z) (y + z) = (x + y) (x + z)

L.H.S. = (x + y) (x + z) (y + z)
= (x + y) (xy + xz + yz + zz)
= (x+y) (xy + xz + yz + z)
= (x + y) (xy + z(x + y + 1))
= (x + y) ( xy + z)
= xxy + xz + xyy + yz
= 0 + xz + xy + yz
= xz + xy + yz

R.H.S. = (x + y) (x + z)
= xx + xz + xy + yz
= 0 + xz + xy + yz
= xz + xy + yz
L.H.S = R.H.S [Proved]