Two sets being equal means that they have the same elements.
So, take an arbitrary xÎ(AÇB)'
Þ xÏAÇB (by definition of complement)
Þ xÏA or xÏB (where or is interpreted in the mathematical sense as meaning either or both)
Þ xÎA' or xÎB' (again using the definition of complement)
Þ xÎA'ÈB'
Therefore, (AÇB)' is a subset of A'ÈB' (ie all elements in the first set are in the second set). To complete the proof, it is necessary to show that all elements in the second set are in the first set, which I'll leave to you (you can pretty much do the above in the reverse order).
So, take an arbitrary xÎ(AÇB)'
Þ xÏAÇB (by definition of complement)
Þ xÏA or xÏB (where or is interpreted in the mathematical sense as meaning either or both)
Þ xÎA' or xÎB' (again using the definition of complement)
Þ xÎA'ÈB'
Therefore, (AÇB)' is a subset of A'ÈB' (ie all elements in the first set are in the second set). To complete the proof, it is necessary to show that all elements in the second set are in the first set, which I'll leave to you (you can pretty much do the above in the reverse order).