Sets

**matsci0000** · July 16th 2009, 09:43 AM

Is there any direct formula to find the complement of (A-B)?

**HallsofIvy** · July 16th 2009, 10:03 AM

Originally Posted by matsci0000

Is there any direct formula to find the complement of (A-B)?

(complement of A) union (A intersect B)

**Grandad** · July 16th 2009, 12:32 PM

Hello matsci0000

Originally Posted by matsci0000

Is there any direct formula to find the complement of (A-B)?

A slightly simpler version of the formula that HallsofIvy has given you is $A' \cup B$ , where $A'$ is the complement of $A$ (which is sometimes written $A^c$ ).

If you know the Laws of Sets, you can prove this as follows:

First, the definition: $A - B = A \cap B'$ (the elements that are in $A$ but not in $B$ )

So $(A-B)' = (A \cap B')'$

$= A' \cup (B')'$ (De Morgan's Law)

$= A' \cup B$ (Involution Law)

Grandad

**HallsofIvy** · July 16th 2009, 04:07 PM

Thanks, Grandad, I didn't take the time to simplify.

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Complement of (A - B)

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