Hello. I'm Professor Von Schmohawk and welcome to Why U.
We have discussed several operations
which create new sets by combining sets in various ways.
The union of sets contains all elements which are members of either set.
The intersection of sets contains only elements which are members of both sets.
The relative complement of A in B contains all elements which are members of B but not A.
And the relative complement of B in A contains all elements which are members of A but not B.
However, there is another possible combination.
We can form a collection of elements which are members of A or B but not both
in other words, the union of A and B without the intersection.
This forms what is called the "symmetric difference of A and B".
The symmetric difference operand is the Greek letter delta.
We now have five different ways to combine two sets to create a new set.
By using these operations to combine more than two sets
we can create interesting combinations.
For example, when we first introduced Venn Diagrams
we showed a diagram which looked like this.
This diagram represents the symmetric difference of three intersecting sets.
To see why, start by taking the symmetric difference of the first two sets A and B.
This operation forms a set with an empty region in the middle
since the symmetric difference excludes intersecting regions.
We then take the symmetric difference
of the set we just formed with the third set C.
This second symmetric difference excludes any regions
which intersect between C and the previous set.
However, in the empty region of the previous set
there is nothing to intersect with set C.
Therefore any part of C which overlaps this empty region will not be excluded.
Now instead of Venn diagrams, let’s do this one more time with three sets of numbers.
Let’s say that set A contains the numbers one, two, three, and four
set B contains the numbers three, four, five, and six
and set C contains two, four, six, and seven.
We start by forming the symmetric difference of the first two sets A and B.
Notice that sets A and B have the elements three and four in common.
Therefore three and four are the intersection of sets A and B.
Since the symmetric difference of two sets
includes all the elements of both sets excluding their intersection
the symmetric difference of A and B consists of all the elements of A and B
except for three and four.
So the symmetric difference of A and B consists of one, two, five, and six.
We now form the symmetric difference of this set with set C.
These two sets have the numbers two and six in common
so their symmetric difference contains all the elements except for two and six.
The symmetric difference of sets A, B, and C
therefore consists of one, four, five, and seven.
Now let’s form this symmetric difference one last time, using numbers and Venn diagrams.
Once again, set A contains the numbers one, two, three, and four
set B contains three, four, five, and six
and set C contains two, four, six, and seven.
We first form the symmetric difference of sets A and B.
These two sets have the elements three and four in common
so their symmetric difference includes all the elements except three and four.
The symmetric difference of A and B
therefore consists of one, two, five, and six.
We now form the symmetric difference of this set with set C.
These two sets have the numbers two and six in common
so their intersection is two and six.
The symmetric difference will therefore contain everything except two and six.
So the symmetric difference of sets A, B, and C
consists of one, four, five, and seven.
Try creating some of your own combinations of sets using the operations we have studied.
Venn diagrams are not only useful but can be beautiful as well.
In the next lecture, we will discuss another useful way to represent sets of numbers
called "interval notation".