Question about Binary relations

**abc512** · September 13th 2009, 09:13 AM

I'm working on binary relation questions and I think I'm just confused about what exactly it means to be a binary relation. One of the questions is

$R[A- B] \supseteq R[A] - R[b]$ and I'm supposed to show that the superset can't be replaced by an equal sign.

All of the B's should be capitalized.

**Plato** · September 13th 2009, 09:16 AM

Originally Posted by abc512

I'm working on binary relation questions and I think I'm just confused about what exactly it means to be a binary relation. One of the questions is

$R[A- B] \supseteq R[A] - R[b]$ and I'm supposed to show that the superset can't be replaced by an equal sign.

All of the B's should be capitalized.

Please define terms.
What are A & B?
What does R[A] mean?

**abc512** · September 13th 2009, 09:23 AM

A and B are sets. R[A] means the binary relation of A.

**Plato** · September 13th 2009, 10:06 AM

Are you sure it is not $R[A- B] {\color{red}\subseteq} R[A] - R[b]$ ?
If not then I do not understand the question.
Because I can give you a counter example to the way you have written it.

**spaceship42** · September 13th 2009, 10:11 AM

Originally Posted by Plato

Are you sure it is not $R[A- B] {\color{red}\subseteq} R[A] - R[b]$ ?
If not then I do not understand the question.
Because I can give you a counter example to the way you have written it.

No, it's definitely the $\supseteq$ . Another problem I have is
$R[A \cap B] \subseteq R[A] \cap R[b]$ and to show that $\subseteq$ can be replaced by =. Would you be able to explain that one better?

**Plato** · September 13th 2009, 10:23 AM

Originally Posted by spaceship42

No, it's definitely the $\supseteq$ . Another problem I have is
$R[A \cap B] \subseteq R[A] \cap R[b]$ and to show that $\subseteq$ can be replaced by =. Would you be able to explain that one better?

OK Look at this example. On $\{1,2,3,4,5,6,7,8\}$
define $R$ as $xRy$ if and only if $x|y$ (x divides y).
Now if $A=\{2,4,6,8\}~\&~B=\{1,2,3,4\}$ then $R[A\backslash B] \subset R[A]\backslash R[ B ]$ .
But $R[A]\backslash R[ B ] \not\subseteq R[A\backslash B]$

So what am I not understanding about the question?

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