# Notation

• Jun 4th 2010, 10:53 AM
Noxide
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
• Jun 4th 2010, 11:05 AM
undefined
Quote:

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.

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

I believe this has the same meaning:

$\displaystyle a \in A \Longrightarrow a \in B$
• Jun 4th 2010, 11:24 AM
Noxide
Quote:

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

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

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
• Jun 4th 2010, 11:36 AM
undefined
Quote:

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:

Quote:

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.