There are lots of operations we can do on sets:

• We can find the UNION of two sets . We find a UNION of two sets by taking every element that is in either set and putting all of these elements together into one big set. To denote the UNION of A and B we write: A B. So the union of A and B is a bigger set that contains both A and B.

Formally we can write: A B = {x| x A or x B}.

(Remember, out loud we would say, “the set of all x’s such that x is an element of A or x is an element of B.”

If an element is in the union of A and B, this means it is in A OR in B.

To see what a union looks like on a Venn diagram, we draw our two sets A and B on a Venn diagram as below:

Then shading in A B in red would look like this:

Let’s look at a few sets and practice taking the union of them.

• Let A={1.1, 1.2, 1.3} and B={1.2, 1.4, 1.6}. If we take every element of A and every element of B and put them all together, we get A B={1.1, 1.2, 1.3, 1.4, 1.6}.
• Let A={0,1,2,3,4} and let B={5,6,7,8,9,10}. Then if we take every element of A and every element of B and put them all together, we get A B={0,1,2,3,4,5,6,7,8,9,10}.

• We can find the INTERSECTION of two sets . We find an INTERSECTION of two sets by taking every element that is in both sets at the same time and putting all of these elements together into one smaller set. To denote the INTERSECTION of A and B we write: A B. So the intersection of A and B is a smaller set that is contained in both A and B.

Formally we can write: A B = {x| x A and x B}.

(Remember, out loud we would say, “the set of all x’s such that x is an element of A and x is an element of B.”

If an element is in the intersection of A and B, this means it is in A AND in B.

To see what an intersection looks like on a Venn diagram, we draw our two sets A and B on a Venn diagram as below:

Then shading in A B in yellow would look like this:

Let’s look at a few sets and practice taking the intersection of them.

• Let A={1.1, 1.2, 1.3} and B={1.2, 1.4, 1.6}. If we take only the elements that are in A and B at the same time and put them all together, we get A B={1.2}.
• Let A={10, 20, 30, 40} and B={10, 15, 20, 25, 30}. If we take only the elements that are in A and B at the same time and put them all together, we get A B={10, 20, 30}.
• Let A={0,1,2,3,4} and let B={5,6,7,8,9,10}. Then if we take only the elements that are in A and B at the same time and put them all together, we get A B= , because there are NO elements in both A and B at the same time. A and B don’t overlap or intersect at all, so their intersection is the null set, or empty set. We have a special name for sets like these:

When the intersection of two sets is empty, we say that the two sets are DISJOINT.

If we drew two disjoint sets on a Venn diagram, they would look like this:

• We can find the COMPLEMENT of a set . We find a COMPLEMENT of a set by taking every element that is in the universal set but not in the set we are looking at and putting all of these elements together into one set. To denote the COMPLEMENT of A we write: A’. So the complement of A is a set that is disjoint with A. If we put A and A’ together, we get the universal set.

Formally we can write: A’ = {x| x U but x A}.

(Remember, out loud we would say, “the set of all x’s such that x is an element of the universal set but x is NOT an element of A.”

If an element is in the complement of A, this means it is NOT in A.

To see what a complement looks like on a Venn diagram, we draw our set A on a Venn diagram as below:

Then we shade in A’ in blue like this:

Likewise, B’ would look like the red shaded area below:

When you are trying to find the COMPLEMENT of a set, don’t let other sets distract you. Notice that when we shade in A’, all we care about is where the set A is on the Venn diagram; we ignore the set B completely for the moment. Likewise, when we shade in B’, we completely ignore the set A, and just shade in all of the area outside the B oval.

Let’s look at a few sets and practice taking the complement of them.

• Let U={1,2,3,4,5} and A={1,3,5}. If we take every element of U that is NOT in A and put them all together, we get A’={2,4}.
• Let U={3.5, 4.5, 5.5} and let A={3.5, 4.5, 5.5}. If we take every element of U that is NOT in A and put them all together, we get A’= .
• Let U={1,2,3,…} and let A={1,2,3,4,5}. If we take every element of U that is NOT in A and put them all together, we get A’={6,7,8,…}.
• Let U=Z and let A= . If we take every element of U that is NOT in A and put them all together, we get A’=U=Z. Because there is nothing in A, the complement of A is the universal set itself. (Remember that a set put together with its complement should give you U.)

• We can find the DIFFERENCE of two sets . We find a DIFFERENCE of two sets by taking every element that is in the first set but not in the second set and putting all of these elements together into one set. To denote the DIFFERENCE of A and be we write: A-B or B-A. A-B is the set of all elements that are in A but NOT in B, and B-A is the set of all elements that are in B but NOT in A. Notice that A-B is always a subset of A and B-A is always a subset of B.

Formally we can write: A-B = {x| x A but x B}.

(Remember, out loud we would say, “the set of all x’s such that x is an element of A but x is NOT an element of B.”

Notice that the complement of A, A’ is the same thing as U-A.

To see what a difference looks like on a Venn diagram, we draw our sets A and B on a Venn diagram as below:

Now we shade in A-B in red like this:

Or we can shade in B-A in blue like this:

Let’s look at a few sets and practice taking the complement of them.

• Let A={1,2,3,4,5} and B={1,3,5}. If we take every element of A that is NOT in B and put them all together, we get A-B={2,4}.
• Let A={3.5, 4.5, 5.5} and let B={3.5, 4.5, 5.5}. If we take every element of A that is NOT in B and put them all together, we get A-B= .
• Let A={1,2,3,…} and let B={1,2,3,4,5}. If we take every element of A that is NOT in B and put them all together, we get A-B={6,7,8,…}.
• Let A=Z and let B= . If we take every element of A that is NOT in B and put them all together, we get A-B=U=Z. Because there is nothing in B, the difference between A and B is A itself.

Notice that B put together with A-B should give you A.

Do you notice the similarities between the complement and the difference between two sets? We could redefine A’ as U-A.