Axiom (The Axiom of Pairs). For every x, y element V there is z element V such t

Question

Axiom (The Axiom of Pairs). For every x, y element V there is z element V such that w element z if and only if w = x or w = y. That is element y element V If x, y element V, then the Axiom of Pairs says that {x, y} is a set. In particular, {x, x} is a set. But {x, x} = {x} (prove this). Taking x = theta, we get that {theta} is a set. Also, {theta}, notequlato theta, since theta element {theta}. We can continue to place more and more brackets around theta and generate more and more distinct sets. This is something that we don't do for classes. If A Bare classes, one of which is proper, then {A, B} is not even a class, much less a set.

Explanation / Answer

x belongs to V then {x,x} is set by axiom of choice.

To prove {x,x} = {x}

It is enough to show that {x,x} is subset of {x} and {x} is subset of {x,x}

For, let an element x from {x,x}

And it is clear that x belongs to {x}

Which implies {x,x} is subset of {x}_______(1)

Now let an element x from {x}

And this x belongs to {x,x}

Which implies that {x} is subset of {x,x} _____(2)

By (1);(2) we can say that both sets are same .

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