# Venn Diagrams Set Theory Explanations

I was hoping someone could explain how these sets map out on the venn diagram. Mainly, i am confused on what the Triangle means in terms of the venn diagram. (The Triangle represents: The symmetric difference between the two sets)

Is there a proof in which i can replace the triangle into unions or intersections?

$A\triangle B$ can be defined either as $(A\cup B)\setminus(A\cap B)$ or as $(A\setminus B)\cup(B\setminus A)$. The two definitions are entirely equivalent, but it’s the latter that explains why $A\triangle B$ is called the symmetric difference of $A$ and $B$: $A\setminus B$ is the difference in one direction, $B\setminus A$ is the difference in the other direction, and taking their union removes any directionality. From an intuitive point of view, however, you might be best off thinking of $A\triangle B$ simply as the set of things belonging to exactly one of $A$ and $B$, just as $A\cap B$ is the set of things belonging to exactly two of $A$ and $B$, and $A\cup B$ is the set of things belonging to at least one of $A$ and $B$.

Now let’s look at $A\triangle(B\cap C)$. $A$ is striped red in the figure below, and $B\cap C$ is solid blue. The points that are in exactly one of those two sets are exactly the points shaded in your picture.

And here they are again, in a modified version of my picture: the remaining blue points are the points that are in $B\cap C$ but not in $A$ (in symbols, in $(B\cap C)\setminus A$), and the remaining red-and-white striped region contains the points that are in $A$ but not in $B\cap C$ (in symbols, in $A\setminus(B\cap C)$). Between the two we have

$$A\triangle(B\cap C)=\underbrace{\Big(A\setminus(B\cap C)\Big)}_{\text{remaining striped region}}\cup\underbrace{\Big((B\cap C)\setminus A\Big)}_{\text{remaining blue region}}\;.$$

Alternatively, we started with the things that were in at least one of the sets $A$ and $B\cap C$ and removed the things that were in both to get the things in exactly one:

$$A\triangle(B\cap C)=\underbrace{\Big(A\cup(B\cap C)\Big)}_{\text{in at least one}}\setminus\underbrace{\Big(A\cap(B\cap C)\Big)}_{\text{in both}}\;.$$

The picture that you already have for $(A\triangle B)\cap(A\triangle C)$ is pretty good. The set $A\triangle B$ is shaded from upper left to lower right (bendwise, if you’re a herald), and the set $A\triangle C$ is shaded from upper right to lower left (bendwise sinister if you’re a herald). The intersection of these two sets consists of those points that are in both sets, so it comprises the regions that are shaded in both directions. In the picture below it’s the blue together with the red-and-white regions.

(The diagrams are a bit crude, but they should help a bit, at least.)

In general, $A\Delta B =(A-B) \cup (B-A)$.
In your Case 1, $A\Delta (B\cap C) = (A-(B\cap C)) \cup ((B\cap C)-A)$, which is clearly shown in the venn diagram. The other case can be worked out in the similar manner.

• Thanks for the quick reply. Just a little confused as to the A&C and A&B situations. Why are C and B are empty. If B&C is not an element of A, How does the venn diagram shade in the areas A&C, and A&B. How do we conclude on the idea that A Union B and A union C is shaded. Commented Mar 15, 2013 at 6:11

$AΔB = (A−B)∪(B−A)$ (or)

$AΔB =(A∪B)-(A∩B)$

If you try to draw the Venn diagrams for both the above statements you will end up with the same venn diagram.

it's really simple , but first let's simplify it ; let's grant numbers to these sets , elements in fact .

Universal set(U):{1,2,3,4,5,6} , A:{1,2} ,B:{1,5,6} ,C:{6,3,4}.

As I am not certain if the image that forms your question is the result , primarily expected or the way that they were organized in a way to know its intersections ,let's start!

A/(B^C) (NOTE:/ IS THE SAME AS THE TRIANGLE AND ^ IS THE SAME AS INTERSECTION ) 1.B^C= the common factor between sets B and C which is 6

1. As we know that B^C is 6 , then this means that A/(B^C) is the same as saying A/6 , so we say : (A-6) U (6-A)= A:{1,2}- B^C ====>{1,2}-{6}====> {1,2}

Now , {6-A}==>6 And as last {1,2} U {6}===>{1,2,6} THE RESULT IS THE SAME IMAGE U POSTED BUT WITH THE NUMBERS, as I am not allowed to post images , imagine this: 3 circles as Mikey Mouse and on his ears you would organize them , from right to left as B , containing 5 , and you will put a 1 between the space in which the ear collides with the other one , and in the other one you will write an A , the name and a 2 , but on his face you will write a 3 and a four , remember his face is set C , co in the part that set B collides with set C you will put a 6 , color that area the A set area and the number 1.

• You have used U for (a) universal set, (b) union of sets, (c) the word "you". Please try to use proper English words. Commented Jul 30, 2015 at 1:03
• I do understand your concerns on the usage of 'u' instead of you , I know that is not a proper word ,in fact changing the Universal set name made me more comfortable , but now I know that no changes are allowed. Commented Jul 30, 2015 at 1:47