Thursday, March 1

practice with demorgan's laws (~x means NOT x)

 ~(~a * ~b)
= ~~a + ~~b
= a + b

 ~(a + ~b)
= ~a * ~~b
= ~a * b

  ~(~a * ~b  +  ~a * ~c  +  b * c * d)
= ~(~a * ~b) * ~(~a * ~c) * ~(b * c * d)
= (~~a + ~~b) * (~~a + ~~c) * (~b + ~c + ~d)
= (a + b) * (a + c) * (~b + ~c + ~d)

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karnaugh maps

example 1:

x   y     out
0   0      0
0   1      1
1   0      1
1   1      1

out =  ~x y  +  x ~y  +  x y

using k-map gives

out =  x + y

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example 2:

x  y  z     out
0  0  0      0
0  0  1	     1
0  1  0	     0
0  1  1	     1
1  0  0	     0
1  0  1	     1
1  1  0	     0
1  1  1	     1

  yz | 00  01  11  10
x    |
-----+----------------
0    | 0   1   1   0
1    | 0   1   1   0

out = z

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example 3:

x  y  z     out
0  0  0      1
0  0  1	     1
0  1  0	     1
0  1  1	     1
1  0  0	     1
1  0  1	     0
1  1  0	     1
1  1  1	     0

  yz | 00  01  11  10
x    |
-----+----------------
0    | 1   1   1   1
1    | 1   0   0   1

out = ~x + ~z

~out = xz
out = ~(xz)
    = ~x + ~z

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example from wikipedia:

a b c d   out
0 0 0 0    0
0 0 0 1	   0
0 0 1 0	   0
0 0 1 1	   0
0 1 0 0	   0
0 1 0 1	   0
0 1 1 0	   1
0 1 1 1	   0
1 0 0 0	   1
1 0 0 1	   1
1 0 1 0	   1
1 0 1 1	   1
1 1 0 0	   1
1 1 0 1	   1
1 1 1 0	   1
1 1 1 1	   0

out = a * ~c  +  a * ~b  +  b * c * ~d     grouping 1s
out = (a + b) * (a + c) * (~b + ~c + ~d)   grouping 0s and using demorgan's law