1. ## groups qustion

i thought that $\displaystyle \supset$ says that the groups cannot be equal
but $\displaystyle \supseteq$ says that they can be equal
so
why if
$\displaystyle A \supset B$ then $\displaystyle A \supseteq B$
??

the one have equality the other doesnt have

2. For any two propositions (i.e., statements that can be either true or false) P and Q, we have that the compound proposition "P and Q" implies P. This is just a fancy way to describe the meaning of the word "and".

Now, let P be $\displaystyle A\supseteq B$ and Q be $\displaystyle A\ne B$. Then $\displaystyle A\supset B$ is P and Q. Therefore, $\displaystyle A\supset B$ implies $\displaystyle A\supseteq B$.

3. Originally Posted by emakarov
For any two propositions (i.e., statements that can be either true or false) P and Q, we have that the compound proposition "P and Q" implies P. This is just a fancy way to describe the meaning of the word "and".

Now, let P be $\displaystyle A\supseteq B$ and Q be $\displaystyle A\ne B$. Then $\displaystyle A\supset B$ is P and Q. Therefore, $\displaystyle A\supset B$ implies $\displaystyle A\supseteq B$.
And to put it another way, $\displaystyle A\supset B$ is a stronger statement than $\displaystyle A\supseteq B$ in the same way that "XYZW is a square" is stronger than "XYZW is a rectangle".

4. I would put it another way.

If $\displaystyle A\subset B$ then $\displaystyle A\subseteq B$ is true.

If $\displaystyle A\subseteq B$ then $\displaystyle A\subset B$ is false.

5. Well we probably have enough ways, but here's another one:

$\displaystyle p \to p\lor q$

is a tautology. Let $\displaystyle \,p$ be $\displaystyle A \supset B$ and $\displaystyle \,q$ be $\displaystyle A = B$.

Of course $\displaystyle p\lor q$ is just $\displaystyle A \supseteq B$ by definition.

6. Originally Posted by Plato
I would put it another way.

If $\displaystyle A\subset B$ then $\displaystyle A\subseteq B$ is true.

If $\displaystyle A\subseteq B$ then $\displaystyle A\subset B$ is false.

I'm guessing you actually meant ( $\displaystyle A \subseteq B\Longrightarrow A\subset B$ ) is false...

Tonio