We are given sets A and B
and asked to determine
the union of A and B,
the intersection of A and B,
as well as A minus B and B minus A.
Let's began by determining A union B
or the union of A and B.
The union of A and B is a set containing all the elements,
which are elements of A or B or both.
Another way to think of a union is to combine the sets
without listing repeated elements more than once.
So analyzing sets A and B,
the elements that are in A or in B or both
are the elements
2, 3,
4, 5,
6, 7, 8.
The elements of 2, 3, 4, 5, 6, 7, 8
are in A or in B or in both.
Next, we have A intersect B or the intersection of A and B.
The intersection of A and B
is the set containing all the elements
which are elements of both A and B.
So analyzing sets A and B.
The elements that are in A and in B
are the elements of 4, 6, and 7.
A intersect B or the intersection of A and B
is the set containing the elements 4, 6, and 7.
Next, we have A minus B.
A minus B is the set of all elements of A
which are not elements of B.
So again, we're looking for the elements
that are in A but not in B.
So 4 is in A, but it's also in B.
5 is in A, and it's not in B.
And therefore, 5 is in A minus B.
6 is in A, but it's also in B.
7 is in A, but it's also in B.
But notice 8 is in A but not in B.
8 is in the set A minus B.
A minus B is a set containing the elements of 5 and 8,
because 5 and 8 are the elements in A,
which are not in B.
And now we want to find B minus A.
B minus A is the set containing the elements of B
that are not in A.
So again we're looking for the elements
that are in B that are not in A.
Notice 2 and 3 are in B but not in A.
4 is in B, but it's also in A.
6 is in B, but it's also in A.
And 7 is in B, but it's also in A,
which means B minus A is a set containing
the elements of 2 and 3.
2 and 3 are the elements of B, which are not in A.
Now, before we go, let's verify this work
by completing the Venn diagram.
To complete the Venn diagram,
we begin by determining
the overlapping region of set A and set B,
which represents the intersection of A and B,
which we know contains the elements of 4, 6, and 7.
So the elements of 4, 6 and 7
go in the intersection of A and B.
And now let's take a look at the set A.
Notice the set A contains the elements 4, 5, 6, 7, 8.
4, 6 and 7 are already in set A,
which means the elements of 5 and 8 are in A but not in B,
and therefore, they go in this region here.
Notice how set A does contain the elements of 4, 5, 6, 7, 8.
And now let's analyze set B.
Notice, we already have 4, 6, 7 in set B,
but the elements of 2 and 3 are also in B.
The elements of 2 and 3 go in this region here,
they're in B but not in A.
And now I can quickly verify our previous work.
A union B,
again, is a set containing the elements
that are in A or B or both.
A union B in the Venn diagram would be
this region here,
where again we're combining the two sets into one set
without listing repeated elements more than once.
Notice A union B
is a set containing the elements 2, 3, 4, 5, 6, 7, 8,
as we found before.
A intersect B or the intersection of A and B
would be the region that's in both set A and set B,
which is this region here, this overlapping region.
Notice A intersect B
is the set containing the elements 4, 6, and 7.
Next, A minus B is the set containing
the elements of A that are not in B.
So here is set A.
The elements in A that are not in B
would be in this region here,
which again is a set containing the elements of 5 and 8.
And then finally,
B minus A is a set containing the elements
that are in B but not in A,
which would be this region here.
B minus A is a set containing the elements of 2 and 3.
The Venn diagram does verify our work is correct.
I hope you found this helpful.