Now this was the explanation on how to solve a prove in my book :
We will prove that the two sets complement (A∩B) and complement(A) ∪ complement(B) are equal by showing that each is a subset of the other.
First,we will show that complement (A∩B) ⊆ complement(A) ∪ complement(B). We do this by showing that if x is in complement (A∩B) ,then it must also be in complement(A) ∪ complement(B). Now suppose that x ∈ complement (A∩B).By the definition of complement,x ∈ complement (A∩B). Using the definition of intersection,we see that the proposition¬((x ∈ A)∧(x ∈ B))is true. By applying De Morgan’s law for propositions, we see that¬(x ∈ A) or¬(x ∈ B). Using the definition of negation of propositions, we have x ∈ A or x ∈ B. Using the definition of the complement of a set, we see that this implies that x ∈ A or x ∈ B. Consequently, by the definition of union, we see that x ∈ complement(A) ∪ complement(B) . We have now shown that complement (A∩B) ⊆ complement(A) ∪ complement(B). Next, we will show that complement(A) ∪ complement(B) ⊆ complement (A∩B).We do this by showing that if x is in complement(A) ∪ complement(B), then it must also be in A∩B.Now suppose that x ∈ A∪B. By the definition of union, we know that x ∈ A or x ∈ B.Using the definition of complement,we see that x ∈ A or x ∈ B.Consequently, the proposition¬(x ∈ A) ∨¬(x ∈ B)is true. By De Morgan’s law for propositions, we conclude that ¬((x ∈ A) ∧(x ∈ B)) is true. By the definition of intersection, it follows that ¬(x ∈ A∩B). We now use the definition of complement to conclude that x ∈ complement (A∩B). This shows that complement(A) ∪ complement(B) ⊆ complement (A∩B). Because we have shown that each set is a subset of the other,the two sets are equal,and the identity is proved.
Why do I have to negate the definition of an intersection to prove that complement (A∩B) = complement(A) ∪ complement(B)?
What does the definition of the intersection got to do with this?
Because this is my first time solving a proof. Am I just suppose to manipulate the equation complement (A∩B) and simplify it by definitions?
Best Answer
If you are asking for an intuitive explanation of why $(A \cap B)^c = (A^c \cup B^c)$, consider a set $X$ containing $A$ and $B$, so that $A^c = X - A$ and $B^c = X - B$. Then we see that any element in $X$ lies in exactly one of the following sets (they are all mutually exclusive):
$$A \cap B$$
$$A^c \cap B$$
$$A \cap B^c$$
$$A^c \cap B^c$$
Then you see that when you ask for what $(A \cap B)^c$ is, you see that it is precisely the union of the three remaining sets: $(A^c \cap B) \cup (A \cap B^c) \cup (A^c \cap B^c)$. As it turns out, we can see that $(A^c \cap B) \cup (A \cap B^c) \cup (A^c \cap B^c) = A^c \cup B^c$.