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.
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).