CODEWITHSUNDEEP
computer-sciencelogical-operatorscpu-architecturediscrete-mathematics
user5412293
user5412293

Reputation:

What is XOR Equivalence

I am trying to see the equivalence of the XOR expression but I cannot. I am trying to express a XOR with only NANDs

 A^B
=AB'+A'B
=(A+B)(AB)'           <= This the transformation I don't understand.
=((A(AB)')'(B(AB)')')'

Thanks a lot for the help :)

Upvotes: 1

Views: 1164

Answers (1)

FriedSaucePots
FriedSaucePots

Reputation: 1352

Let's go backwards with more steps

(A+B)(AB)'

<=>A(AB)' + B(AB)' (distributivity)

<=>A(A'+B') + B(A'+B') (demorgan on (AB)')

<=>AA' + AB' + BA' + BB' (distributivity)

Since AA' and BB' evaluate to false (i.e. T AND F or F AND T result in False), they can be removed from the chained or-condition leaving (False OR X <=> X)

AB' + A'B

So back to the original . . .

1) they start with the AB' + A'B

2) added the terms AA' and BB' giving AA' + AB' + BA' + BB'

3) factored out (A'+B') leaving A(A' + B') + B(A' + B')

4) Rearranged to (A+B)(A'+B')

4) Finally demogran law to result in (A+B)(AB)'

Upvotes: 3

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