<s>[INST] express the union and intersections of the circles How to express  the union  and intersections of the circles $ \ x^2+(y+1)^2=9$ and $x^2+(y-1)^2=9 \ $ in terms of set comprehension notation. 
Answer:
Let , 
$ A=\{(x,y): \ x^2+(y+1)^2=9 \} \ \ and \ \ B=\{(x,y): \ x^2+(y-1)^2=9 \} \ $
Thus,
$A \cup B \ =\{(x,y): x=\pm \sqrt{9-(y+1)^2} \ \cup \ x=\pm \sqrt{9-(y-1)^2} \} $
Am I right ?
If there is more suitable way ?
 [/INST] $x=\pm \sqrt{9-(y+1)^2} \ \cup \ x=\pm \sqrt{9-(y-1)^2}$
is meaningless, because the $\cup$ operator is an operator on sets, not equations. If anything, it should say
$x=\pm \sqrt{9-(y+1)^2} \ \lor \ x=\pm \sqrt{9-(y-1)^2}$
since $\lor$ is the symbol for "or".

Other than that, I see no easy way to describe the set, except maybe just lose the square roots and $\pm$ by writing
$$A\cup B=\{(x,y): x^2+(y+1)^2=9\lor x^2+(y-1)^2=9\}$$
or, shorter, but less informative:
$$A\cup B = \{(x,y): x^2+(y\pm 1)^2=9\}$$
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