Thread: venn diagram

Is the venn diagram for $\displaystyle (A - (A \cap B)) \cup (B - (A \cap B)) $, the same as $\displaystyle (A-B) \cup (B-A)$

where as $\displaystyle A \cap B \neq \varnothing$

Originally Posted by Discrete

Is the venn diagram for $\displaystyle (A - (A \cap B)) \cup (B - (A \cap B)) $, the same as $\displaystyle (A-B) \cup (B-A)$

where as $\displaystyle A \cap B \neq \varnothing$

yes, both will look like this: (the shaded area is the desired set)

Originally Posted by Discrete

Is the venn diagram for $\displaystyle (A - (A \cap B)) \cup (B - (A \cap B)) $, the same as $\displaystyle (A-B) \cup (B-A)$
where as $\displaystyle A \cap B \neq \varnothing$

Yes and we do not even need $\displaystyle A \cap B \neq \varnothing$
This is known as the symmetric difference.

Originally Posted by Discrete

Is the venn diagram for $\displaystyle (A - (A \cap B)) \cup (B - (A \cap B)) $, the same as $\displaystyle (A-B) \cup (B-A)$

where as $\displaystyle A \cap B \neq \varnothing$

The reason is :
$\displaystyle (A-(A\cap B))\cup(B - (A \cap B)) $ = $\displaystyle ((A-A)\cup(A-B))\cup((B-A)\cup(B-B)) $ = $\displaystyle \varnothing\cup (A-B)\cup (B-A)\cup\varnothing = (A-B) \cup (B-A)$

Jhevon, what software did you use to draw that Venn diagram?

Originally Posted by rualin

Jhevon, what software did you use to draw that Venn diagram?

regular, good-old-fashioned MS-Paint. I use that for pretty much all my diagrams besides graphs.

Hello, Discrete!

Here's another proof . . .

Prove: .$\displaystyle [A - (A \cap B)] \cup [B - (A \cap B)] \;=\;(A-B) \cup (B-A)$

We need (among other definitions and properties):

. . [1] .$\displaystyle P - Q \;=\;P \cap\, \overline{Q}$ . (Def. of set subtraction)
. . [2] .$\displaystyle P \cap \overline P \:=\:f$
. . [3] .$\displaystyle P \cup f \:=\:P$


We have: .$\displaystyle [A - (A \cap B)] \cup [B - (A \cap B)]$

. . . . . $\displaystyle = \;\left[A \cap (\overline{A \cap B})\right] \cup \left[B \cap \left(\overline{A \cap B}\right)\right]$ . . . . . . . . . . . . . . . [1]

. . . . . $\displaystyle = \;\left[A \cap \left(\overline A \cup \overline B\right)\right] \cup \left[B \cap \left(\overline A \cup \overline B\right)\right] $ . . . . . . . . . . . DeMorgan's Law

. . . . . $\displaystyle = \;\left[\left(A \cap \overline A\right) \cup \left(A \cap \overline B\right)\right] \cup \left[\left(B \cap \overline A\right) \cup \left(B \cap \overline B\right)\right]$ . . Distributive Property

. . . . . $\displaystyle = \;\left[f \cup \left(A \cap \overline B\right)\right] \cup \left[\left(B \cap \overline A\right) \cup f\right]$ . . . . . . . . . . . . . . . [2]

. . . . . $\displaystyle = \;\left(A \cap \overline B\right) \cup \left(B \cap \overline A\right)$ . . . . . . . . . . . . . . . . . . . . . . [3]

. . . . . $\displaystyle = \;(A - B) \cup (B - A)$ . . . . . . . . . . . . . . . . . . . . . . [1]