Expert: Paul Klarreich - 9/16/2009

Question
Let A,B be a nonempty proper subsets of R and x not in A. Prove ((-∞,x)U(x,∞))∩A=A

Answer
Questioner: Stephanie
Country: United States
Category: Advanced Math
Private: No
Subject: Analysis
Question: Let A,B be a nonempty proper subsets of R and x not in A. Prove ((-∞,x)U(x,∞))∩A=A
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I am going to write:
((- inf, x) + (x, +inf)) * A = A
for your: ((-∞,x)U(x,∞))∩A=A

because I can't make those things.

So my notation is:
 +   is union,
 *   is intersection.
 /=  is 'not equal'
 x in A means  'x belongs to A'

Now to show two sets are equal, take some c that is in the left side and show that it is in the right side, then do the vice versa.(a dance craze of the 50's?)

Now the first set on the left: ((- inf, x) + (x, +inf))
is just all reals except x, so we could write:

{ all reals /= x } * A = A  as your statement to prove.

Part I: Suppose  c is in left side.  If c is in an intersection, then it must be in both of the sets, therefore in A.  
First part proved.

Part II: Suppose  c is in right side.  Then c in A.
Claim:  c is in both sets in the left side.  Obviously c is in A.

Is c in { all reals /= x }?  Yes, because (you said) x not in A.  Therefore c is a real that /= x.  So c is in both sets, therefore in the intersection.
Second part proved.

ALTERNATIVE SECOND PART:

Suppose  c is not in the left side.  Then
AT LEAST ONE OF THESE IS TRUE:

1. c not in A.
2. c not in  { all reals /= x }, which means  c = x, which means c not in A.

Either way,  c not in A.  So  c not in (right side).