Now let’s look at a couple of new terms and concepts. What is an intersection? Probably you think of a place where 2 or more things cross each other. Take for example a street intersection. In a sense, this is where the streets overlap. It’s a common area shared by all the streets involved in the intersection.
Look at the 2 sets above. V = {a, e, i, o u, y, w} and C = {b, c, d, f, g, h, j, k, l, m, n, p, q, r, s, t, v, w, x, y, z}. Where do these 2 sets intersect, if anywhere? Another way to ask this question is: Do these 2 sets share any common elements? If you look closely, you’ll see that V & C do have 2 common elements: y, and w. This is called the intersection of V & C.
We can depict this situation in a Venn diagram as above. Written with the symbol
for intersection, , we write V
C = {y, w}. In words, this means
the set, V,
intersected with the set, C, is the set of elements y and w. If we name the
intersection B, so y 
B and w
B, then we can write V
C = B.
We’re beginning to use a lot of symbols here, which you might be used to, but keep practicing because it’s takes some time to learn this new language.
Here’s another definition. The union of 2 sets is the set of elements which
belong to one or the other or both sets. In other words, the union contains all
the elements of the 2 sets regardless of where they exist. E.g., the union of V
and C above can be written as V
C and it contains the entire alphabet.
Now let’s take a look at a simple set of 3 elements. S = {x, y, z}. How many subsets can you form from this set?
Recall that we can use the math symbol,
, to mean is a subset of. If you’re
using a keyboard and can’t write
, then just write it out in words instead.
Well, let’s just try listing the subsets we find. A set can contain just 1 element, so we can list those: {x},{y}, and {z}. Likewise, a set could contain only 2 elements, so let’s list those: {x, y}, {x, z}, and {y, z}. Notice that we don’t need to list {y, x} {z, x}, and {z, y} because they really are the same sets as {x, y}, {x, z}, and {y, z}, respectively. In other words, the ordering of elements within a set does not matter. You can think of a set as being like a bowl of coins. The total value of the coins remains the same no matter how much the coins are mixed up.
Finally, we have 1 set of 3 elements: {x, y, z}. In other words, the set is a
subset of itself. Actually, if you look at the symbol for subset,
, you’ll see
it looks a bit like equals sign = except that the top bar is an
instead.
So how many sets do we have altogether? Yup, 7. But actually there is one more missing, and this is a tricky one. We also can count the empty set which is the set containing nothing! This is called the null set. So we have a total of 8 subsets.
Practice
Try answering the below questions. Click on ANSWER drop-down boxes to check your answers.