A contradiction arising due to set-builder form.

Question

I have noticed a kind of contradiction arising due to set-builder form involving universal set$(U)$ while I was performing operations on set theory.

Here it is -

Let us consider an arbitrary non-empty set $B$. Then, $B = \{x : x \in B\} = \{x : x \in B \wedge x \in U\} = \{x : x \in B \cap U\} = \{x : x \in U\} = U$

But this is contradiction as $B \not= U$. Instead $B \subseteq U$.

Is it really a contradiction or my misconception ?

A detailed explanation would be helpful.

Answer 1

As has been explained in the comments (and if one of the original commenters adds an answer, I'll delete this one), the error is not a subtle logical issue but rather an intersection calculation.

You ask,

Why can't we consider $B\cap U=U$ instead of $B\cap U=B$?

Well, the answer is: the first one is not necessarily true given our hypotheses on $B$ and $U$, while the second one is necessarily true given those hypotheses. I suspect the special character of $U$ (= the universal set) is making this more confusing than it should be.

Forgetting about universal sets, remember that whenever $X\subseteq Y$ we have $X\cap Y=X$, but we do not necessarily have $X\cap Y=Y$ (indeed, we won't unless $X=Y$). For example, take $X=\{1\}, Y=\{1,2\}$; we have $$\{1\}\cap \{1,2\}=\{1\}\quad\mbox{but}\quad\{1\}\cap\{1,2\}\not=\{1,2\}.$$

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Of course the universal set does pose serious issues, and the usual choice of axioms for set theory in fact prohibits such an object (although some other set theories allow it at the cost of weakening other axioms); but that's irrelevant to the particular issue here, which is really just an issue with subsets and intersections.

Answer 2

The mistake might originate from a lack of familiarity with the propositional constant "t".

The sentence " x belongs to U " is a tautology , whatever x might be ( since, the defintion of the universal set means that any element you are considering belongs to the universal set).

Now, in propositional logic there is a propositional constant denoted by " t" which is defined as : " truth" , "the tautology" , the proposition that is equivalent to any tautology.

Remark : there is also a propositional constant called : " f" " false" ( and that is defined as : the proposition that is equivalent to any antilogy.

Now, propositional logic tells you that, for any proposition P :

(P& t) is equivalent to P.

This is " identity law for tautologies" ( http://mathonline.wikidot.com/the-laws-of-propositional-logic , see also Lipschutz, Schaum's outline of set theory, at archive.org)

Explanation : in order a conjunction to be true, both propositions have to be true; since t is true in all possible case ( being a tautology), everything depends on the truth value of P: if P is true the whole conjunction in true, if P is false, the whole conjunction is false.

With this in mind, what is the set : B Inter U ?

B Inter U = { x | x belongs to B & x belongs to U }

But, as we said, the sentence " x belongs to U " is equivalent to t .

So,

B Inter U

= { x | x belongs to B & x belongs to U }

= { x | x belongs to B & t }

= { x | x belongs to B}

= B