# Thread: Notation

1. ## Notation

I am trying to say: Every a in A is in B

Is the following notation correct?

[(universal quantifier) a (epsilon) A] (epsilon) B

2. Originally Posted by Noxide
I am trying to say: Every a in A is in B

Is the following notation correct?

[(universal quantifier) a (epsilon) A] (epsilon) B
I could be wrong but I think what you wrote would not be considered correct use of notation.

I would instead write:

$\displaystyle \forall\ a \in A: a \in B$

I believe this has the same meaning:

$\displaystyle a \in A \Longrightarrow a \in B$

3. Originally Posted by undefined
I could be wrong but I think what you wrote would not be considered correct use of notation.

I would instead write:

$\displaystyle \forall\ a \in A: a \in B$

I believe this has the same meaning:

$\displaystyle a \in A \Longrightarrow a \in B$

$\displaystyle \forall\ a \in A: a \in B$

does ":" mean such that?

If you're interested I'm just talking about the case where A is a subset of B

4. Originally Posted by Noxide
$\displaystyle \forall\ a \in A: a \in B$

does ":" mean such that?

If you're interested I'm just talking about the case where A is a subset of B
You could read ":" aloud by simply pausing: "For all a in A, a in B." Or you could read it as "it is true that."

Here's another way.

$\displaystyle A \subseteq B$ if and only if $\displaystyle \forall x(x\in A \Rightarrow x\in B)$

Edit: upon reflection, what I wrote above:

Originally Posted by undefined
I believe this has the same meaning:

$\displaystyle a \in A \Longrightarrow a \in B$
isn't precise enough, because $\displaystyle a$ is unspecified, thus the expression is not a sentence/proposition in the language (it cannot be assigned a truth value); we would still need the $\displaystyle \forall$ symbol, like I did in this post.