# Set theory

1. Nov 8, 2014

### Panphobia

1. The problem statement, all variables and given/known data
Set A has twice the number of elements as Set B, 1/3 of the elements of Set A are the same as in Set B, the union of A and B is 42, what is the intersection?

3. The attempt at a solution

This was one of my exam questions, and I just want to see what the correct answer was. What I tried to do was
use the inclusion exclusion principle so

|AUB| = |A| + |B| - (1/3)*|A|
42 = |A| + 1/2|A| - (1/3)|A|
42 = (1/6)|A| + (3/6)*|A| - (2/6)*|A|
42 = (1/3)|A|

And 42 is the intersection, but that makes absolutely no sense, can anyone show me the correct way to get the answer?

2. Nov 8, 2014

### phinds

Draw a Venn diagram using Kn in each section (K different for each section). Answer of 12 falls out trivially.

3. Nov 8, 2014

### Ray Vickson

$$42 = |A \cup B| = |A| + |B| - |A \cap B| = 2|B| + |B| - (1/3)\underbrace{(2 |B|)}_{|A|} = (3 - 2/3) |B| = (7/3) |B|$$.

4. Nov 8, 2014