How do you construct proofs for set unions?

[fruitjunkie]

union proof due at midnight!

(∃y)(∀x)(x ∈ y) ↔ (x ∈ a ∨ x ∈ b))
How do you prove this??

 

[micromass]

In ZF, that is an axiom. But I think you don't really mean that. So could you elaborate??

 

[fruitjunkie]

it is a lemma that my professor asked us to prove.

 

[micromass]

So, what did you try already??

 

[fruitjunkie]

i actually don't know how to construct proofs..

 

[mathwonk]

i don't get it. the letters a and b seem to be unbound variables, hence meaningless. so there is no statement here. what gives??

 

[fruitjunkie]

oops, i forgot the beginning: Given sets a and b, there is a set containing exactly the elements from a and b:

 

[HallsofIvy]

It's well past midnight but that is just as well. If you honestly do not "how to construct proofs" your professor needs to know that so he/she can teach you. If you get someone else to do the problem for you, the professor might think you already know how and not discover the mistake until an exam!

In any case, I doubt that anyone here could give a proof that you would understand without knowing what basics info you have about sets. As micromass said, in ZF, that's an axiom. What "axioms" or operations do you have to work with?

 

[AdamFiddler]

It's well past midnight but that is just as well. If you honestly do not "how to construct proofs" your professor needs to know that so he/she can teach you. If you get someone else to do the problem for you, the professor might think you already know how and not discover the mistake until an exam!

In any case, I doubt that anyone here could give a proof that you would understand without knowing what basics info you have about sets. As micromass said, in ZF, that's an axiom. What "axioms" or operations do you have to work with?

The "union axiom" can be derived from the sum axiom and the pairing axiom, as well the proper definitions attached to both of these.