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Do 'sum-of-products' and 'product-of-sums' represent the same function? Does it have be the same expression or not? In case it is different, what does it mean?

Context: I've just made a Karnaugh map and the sum of products give me the same function as the product of sums. Also, the exercise asks to compare the expressions and in case they are not the same I had to explain why. So I was confused.

Diego Vega
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  • Why do you feel they must be the same? – jameselmore Apr 26 '15 at 15:52
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    It only require middle school algebra. Which part do you have problem? Understanding the two phrase or doing middle school algebra? – MonkeyKing Apr 26 '15 at 15:58
  • Just wanted to be sure. I've just made a Karnaugh map and the sum of products give me the same function as the product of sums. Also, the exercise asks to compare the expressions and in case they are not the same I had to explain why. So I just was confused, sorry for asking. – Diego Vega Apr 26 '15 at 16:25

1 Answers1

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If I interpret correctly your statement, you'd like that, in a Boolean algebra, $$ ab+cd=(a+b)(c+d) $$ for all $a,b,c,d$.

However, this isn't true: take $a=0$, $b=1$, $c=0$ and $d=1$. Then $$ 0\cdot1+0\cdot1=0+0=0 $$ while $$ (0+1)(0+1)=1\cdot 1=1 $$

What's true is that $$ (ab+cd)'=(a'+b')(c'+d') $$ (by de Morgan's laws).

egreg
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