# Thread: Converting Sets to Set Builder Notation

1. ## Converting Sets to Set Builder Notation

I have a problem I need to do, and I was wondering what the method is to convert set notation to set builder notation. The problem is:

Show that if A, B, and C are sets than

A $\displaystyle \cup$ B $\displaystyle \cup$ C = A + B + C - (A $\displaystyle \cap$ B) - (A $\displaystyle \cap$ C) - (B $\displaystyle \cap$ C) + (A $\displaystyle \cap$ B $\displaystyle \cap$ C)

If there is a better method than converting to set builder notation, let me know what it is (Although I would still like to know a general method from converting from this notation to a set builder notation).

Thanks,
James

2. Originally Posted by menmonmay
I have a problem I need to do, and I was wondering what the method is to convert set notation to set builder notation. The problem is:
Show that if A, B, and C are sets than
A $\displaystyle \cup$ B $\displaystyle \cup$ C = A + B + C - (A $\displaystyle \cap$ B) - (A $\displaystyle \cap$ C) - (B $\displaystyle \cap$ C) + (A $\displaystyle \cap$ B $\displaystyle \cap$ C)

If there is a better method than converting to set builder notation, let me know what it is
It appears as if you are dealing with the number of elements in a set.
That is denoted by $\displaystyle |A|$ or perhaps $\displaystyle \#(A)$.
Thus your formula would be: $\displaystyle |A\cup B\cup C|=|A|+|B|+|C|-|A\cap B|-|A\cap C| -|B\cap C|+|A\cap B\cap C|$

BTW: The LaTex for that is $$|A\cup B\cup C|=|A|+|B|+|C|-|A\cap B|-|A\cap C| -|B\cap C|+|A\cap B\cap C|$$.