Homework Answers: Boolean Algebra

 

  1.  
    A    		  
    B    		  
    C    		  
    B    		  
    C    		  
    BC    		  
    BC    		  
    BC + BC    		  
    A · (BC + BC)
    0 0 0 1 1 0 0 0 0
    0 0 1 1 0 0 1 1 0
    0 1 0 0 1 1 0 1 0
    011000000
    1 0 0 1 1 0 0 0 0
    101100111
    1 1 0 0 1 1 0 1 1
    1 1 1 0 0 0 0 0 0

    2.  

  2.  
    A    		  
    B    		  
    C    		  
    A    		  
    B    		  
    A + B    		  
    A + C    		  
    A + B    		  
    (A + B)·(A + C)·(A + B)
    0 0 0 1 1 0 0 1 0
    001100110
    0 1 0 1 0 1 0 1 0
    011101111
    1 0 0 0 1 1 1 1 1
    101011111
    1 1 0 0 0 1 1 0 0
    1 1 1 0 0 1 1 0 0

    3.  

  3.  
    1    		  
    P    		  
    1 · P
    1 0 0
    1 1 1

    4. The values for P and 1 · P match in both rows, so the identity is true.

  4.  
    P    		  
    Q    		  
    P + Q    		  
    P + Q    		  
    P    		  
    Q    		  
    P · Q
    0 0 0 1 1 1 1
    0 1 1 0 1 0 0
    1 0 1 0 0 1 0
    1 1 1 0 0 0 0

    5. The values for P + Q and P · Q match in every row, thus the identity is true.

  5.  
    P    		  
    Q    		  
    R    		  
    P + Q    		  
    Q + R    		  
    P + (Q + R)    		  
    (P + Q) + R
    0 0 0 0 0 0 0
    0010111
    0 1 0 1 1 1 1
    0111111
    1 0 0 1 0 1 1
    1 0 1 1 1 1 1
    1 1 0 1 1 1 1
    1 1 1 1 1 1 1

    6. The values for P + (Q + R) and (P + Q) + R match in every row thus the identity is true.

  6.  
    A    		  
    B    		  
    A    		  
    B    		  
    A + B    		  
    A + B    		  
    (A + B) · (A + B)
    0 0 1 1 1 1 1
    0 1 1 0 1 1 1
    1 0 0 1 1 0 0
    1 1 0 0 0 1 0

    7. The values for (A + B) · (A + B) and A match in every row thus the identity is true.

  7.  
    A    		  
    B    		  
    A    		  
    A · B    		  
    A + (A · B)    		  
    A + B
    0 0 1 0 0 0
    0 1 1 1 1 1
    1 0 0 0 1 1
    1 1 0 0 1 1

    8. The values for A + (A · B) and A + B match in every row and thus the identity is true.

  8. A · (A + B) = AA + AB = AB

  9. T · U · V + X · Y + Y = TUV + Y

  10. (B · E + C + F) · C = BCE + C + CF = BCE + C = C

  11. A · B + A · C + B · A = AB + AC

  12. P + Q + P · Q = P + Q + P + Q = 1

  13. (X + Y)·(X + Y) = X + X·Y + X·Y + Y·Y = X

  14. W · (W X Y Z) = W · (W + X + Y + Z) = W + W X + W Y + W Z = W + W Y + W Z = W + W Z = W

  15. ~A + D · B + ~C · ~C + D = AD · BC · CD = ABCD

  16. (A · ~B) · (~A · C) = (A · ~B) + (~A · C) = A + B + A + C = 1 + B + C = 1 + C = 1

  17. (X + Y)(X + Y)(X + Z) = (XX + XY + XY + YY) ( X + Z)
        = (X + X Y + X Y) (X + Z)
        = X X + X Z + X X Y + X Y Z + X X Y + X Y Z
        = X Z + X Y Z + X Y Z
        = X Z

  18. It is false. You can prove it using a truth table. It's harder, but you can also prove it using the relation a XOR b = a b + a b.

  19. F = x y z + x y z + x y z + x y z + x y z

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