MHB Can Set Union Have an Additive Inverse Like Real Numbers?

[paulmdrdo1]

1.show that there is no axiom for set union that correspond to "Existence of additive inverses" for real numbers, by demonstrating that in general it is impossible to find a set X such that $A\cup X=\emptyset$. what is the only set $\emptyset$ which possesses an inverse in this sense?

2. show that the operation of subtraction is not commutative,that is, it is possible to find real numbers a and b such that $b-a\not = a-b$. what can be said about a and b if $b-a=a-b?$

what to do? i don't understand what question 1 is asking.

 

[Evgeny.Makarov]

what to do? i don't understand what question 1 is asking.

Question 1 asks you to find a set $A$ such that $A\cup X\ne\emptyset$ for all sets $X$. Such $A$ is a counterexample to the property

For every $A$ there exists an $X$ such that $A\cup X=\emptyset$ (*)

which is an analog of the "Existence of additive inverses" for real numbers. It also asks to find a unique set $A$ for which (*) is true.

 

[Deveno]

If A is non-empty (for concreteness, let A = {a}), then A U X is also non-empty, because no matter what X is, A U X contains at least the element a.

Thus if A U X = Ø (for some X) it must be the case that A = Ø. What must X be, here?

 

[paulmdrdo1]

X must also be empty. am i right? and evegenymakarov what does this symbol mean (*)?

 

[Evgeny.Makarov]

X must also be empty. am i right?

Yes.

evegenymakarov what does this symbol mean (*)?

I denoted the statement "For every $A$ there exists an $X$ such that $A\cup X=\emptyset$" by (*) in order to refer to it later. This is often done in math texts. The label like (*) or (1) is usually located near the right page margin.