1. ## Very simple question

For the notation $\displaystyle x \in A$,

Say $\displaystyle x \in A \Rightarrow x \in A \cup B$

Does $\displaystyle x \in A$ denote all element $\displaystyle x$ of $\displaystyle A$?

Take a different example:

Say $\displaystyle x \in A \Rightarrow x \in A - B$

Does $\displaystyle x \in A$ denote only some element $\displaystyle x$ of $\displaystyle A$?

How do you decide whether it's some $\displaystyle x$ or all $\displaystyle x$?

2. By itself, the statement $\displaystyle x \in A \Rightarrow x \in A \cup B$ is meaningless.

However, the statement $\displaystyle \forall x, x \in A \Rightarrow x \in A \cup B$ does make sense.

Implications such as this only make sense with universal quantifiers.

3. Originally Posted by novice
For the notation $\displaystyle x \in A$,
Say $\displaystyle x \in A \Rightarrow x \in A \cup B$
Does $\displaystyle x \in A$ denote all element $\displaystyle x$ of $\displaystyle A$?
Take a different example:
Say $\displaystyle x \in A \Rightarrow x \in A - B$
Does $\displaystyle x \in A$ denote only some element $\displaystyle x$ of $\displaystyle A$?
How do you decide whether it's some $\displaystyle x$ or all $\displaystyle x$?
The set of all elements in $\displaystyle A$ is $\displaystyle \{x:x\in A\}$

The set $\displaystyle \{x:x\in (A\setminus B)\}$ is the set of all $\displaystyle x$ belonging to $\displaystyle A$ but not to $\displaystyle B$.

Now what is the question?

4. Originally Posted by icemanfan
By itself, the statement $\displaystyle x \in A \Rightarrow x \in A \cup B$ is meaningless.
It is not meaningless. It says "If x is in A then x is in A union B".
That is perfectly good well formed English sentence. And it is true.

5. Originally Posted by Plato
The set of all elements in $\displaystyle A$ is $\displaystyle \{x:x\in A\}$

The set $\displaystyle \{x:x\in (A\setminus B)\}$ is the set of all $\displaystyle x$ belonging to $\displaystyle A$ but not to $\displaystyle B$.

Now what is the question?
Plato,

You have answered my question pointedly by showing the set builder notation.

Thanks.