# sets

#### solakis

##### Active member
In the book: SET THEORY AND LOGIC By ROBERT S.STOLL in page 19 the following theorem ,No 5.2 in the book ,is given:

If,for all A, AUB=A ,then B=0

IS that true or false

If false give a counter example

If true give a proof

##### Well-known member
This is true.
To prove the same we have $A \cup B = A$ iff $B \subseteq A$

Let us take 2 sets $A_1,A_2$ which are disjoint and because it is true for every set $A_1 \cup B = A_1$ so $B \subseteq A_1$

and $A_2 \cup B = A_2$ so $B \subseteq A_2$

so from above 2 we have

$B \subseteq A_1 \cap A_2$

because $A_1,A_2$ are disjoint sets so we have $A_1 \cap A_2= \emptyset$

so $B = \emptyset$

• lfdahl, topsquark and solakis

#### solakis

##### Active member
Thanks ..........Let $$\displaystyle \forall A[A\cup B=A]$$...........................................................1

put $$\displaystyle A=0$$ and 1 becomes $$\displaystyle 0\cup B=0$$

And $$\displaystyle B=0$$ since $$\displaystyle 0\cup B=B$$

Note 0 is the empty set

However somebody sujested the following counter example:

A={1,2,3}...........B={1,2} so we have :$$\displaystyle A\cup B=A$$ and $$\displaystyle \neg(B=0)$$