/sci/ - Science & Math

What are some ways of deriving a contradiction in an equality between two numbers (or sets) that you know are not equal?

like if you have x = y, but you know x is even and y is odd, you can do 2k = 2l+1 and show 1 = 2(k-l) so it implies 1 is even.

anything else like that?

Is this a valid proof?

Let XUY = X for any X. Prove Y = [math]\emptyset [/math]
Suppose to the contrary that Y [math] \neq \emptyset [/math]. Define X such that X [math] \cap [/math] Y = [math] \emptyset [/math]. Then [math] (\exists x)(x \in Y) \Rightarrow a \in Y \Rightarrow a \in X \cup Y \Rightarrow a \in X \ (sub). \\ Then \ a \in X \wedge a \in Y \Rightarrow a \in X\cap Y \ \Rightarrow (\exists x)(x \in X\cap Y) \Rightarrow X\cap Y \neq \emptyset. \\ A \ contradiction. [/math]

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