1. ## groups qustion

i thought that $\supset$ says that the groups cannot be equal
but $\supseteq$ says that they can be equal
so
why if
$A \supset B$ then $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 $A\supseteq B$ and Q be $A\ne B$. Then $A\supset B$ is P and Q. Therefore, $A\supset B$ implies $A\supseteq B$.

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

4. I would put it another way.

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

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

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

$p \to p\lor q$

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

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

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

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

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

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

Tonio