1
Using Venn Diagrams to demonstrate the Distributive properties
of the union, intersection and set complement operators
Combining the union and intersection operators
Property:
Union is distributive over Intersection
That is, given any sets
A
,
B
and
C
,
A
∪
(
B
∩
C
) = (
A
∪
B
)
∩
(
A
∪
C
)
To demonstrate this using Venn diagrams, we work our way up to diagrams depicting the sets cor
responding to each of the left and right sides of this equation, to see that we get the exact same set
for each.
Step 1
Left Hand Side set:
We need to draw a Venn diagram showing the set
A
∪
(
B
∩
C
).
We start by drawing, separately, the sets
A
and (
B
∩
C
), and then use those diagrams to draw the
union of those sets. When we take the
union
of sets shaded in different Venn diagrams, we use (i.e.,
shade)
all
parts of the final diagram which are shaded in
either
of the two diagrams.
A
C
B
A
C
B
A
C
B
Figure 1(a):
Figure 1(b):
Figure 1(c)
Set A.
The subset
B
∩
C
.
A
∪
(
B
∩
C
) is everything which
is shaded in 1(a) or 1(b).
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 Winter '11
 AndreyMinchenko
 Math, Set Theory, Sets, Symmetric difference

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