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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?

enter image description here

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3 Answers 3

$A\triangle B$ 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.

enter image description here

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:

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

enter image description here

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.

enter image description here

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

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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.

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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. –  MatthewL Mar 15 '13 at 6:11
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$AΔB = (A−B)∪(B−A)$ (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.

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