# Properties of real numbers

#### paulmdrdo

##### Active member
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.

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#### Evgeny.Makarov

##### Well-known member
MHB Math Scholar
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

##### Well-known member
MHB Math Scholar
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?

#### paulmdrdo

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

#### Evgeny.Makarov

##### Well-known member
MHB Math Scholar
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.