We can draw a Venn diagram to see these two sets of students.

We need to find the overlap between these two sets. To find that, add up the total number of students who are taking history and the total number of students who are taking calculus.

Notice that we have more students this way than the total number who were polled. That is because the students who are taking history AND calculus have been double counted. Subtract the total number of students polled to find out how many students were counted twice.

The notation  stands for "union," which refers to everything that is in either set.  refers to the group of students taking either Calculus or Spanish (everyone on this diagram except those taking only Biology). From the diagram, Patrick and Ashley are the only students taking neither Calculus nor Spanish, so Patrick is the correct answer.

We can draw a Venn diagram of these students.

Drawn this way, there are more students on the Venn diagram than we have.

This is because some of the students play both sports and should be in the overlap on the Venn diagram. To find the number of students in the overlap, subtract the total number of students given from the number on the diagram.

This represents the number of students who were counted twice, or the number in the overlap.

We can redraw the correct Venn diagram with this number.

The notation  stands for "A union C," which refers to everything that is in either set  or set .

When we add the numbers together, we get:

A Venn diagram can help us determine the total number of students in the class.

First, we must calculate the number of students who have ONLY cats or ONLY dogs. First, for cats, 15 students have cats, and 5 students have both cats and dogs.

Ten students have only cats.

For dogs, 12 students have dogs, and 5 students have both cats and dogs.

Seven students have only dogs. 

Using this information, we can fill in the Venn diagram.

This diagram shows the 10 students with only cats, the 7 students with only dogs, the 5 students with both, and the 8 students with neither.  Adding up the numbers will give us the total number of students.

 means the intersection of the sets  and , the only things in the intersection are elementes that are in BOTH of the sets. Since there is nothing shared by the two sets (no elements are in both), the interesction is the emptyset, or: 

 is the union of the sets  and  which is the set that contains anything that is in either set. Thus, the total of  is every element that is in one of the two sets, and so  

Draw a Venn diagram with three subsets:  Math, Science, and English.  Start in the center with students that like all three subjects.  Next, look at students that liked two subjects.  Be sure to subtract out the ones already counted in the middle.  Then, look at the students that only like one subject.  Be sure to subtract out the students already accounted for.  Once all of the subsets are filled, look at those students who don’t like any of these subjects.  To find the students who don’t like any of these subjects add all of the students who like at least one subject from the total number of students surveyed, which is 50. 

M = math

S = science

E = English

M∩S∩E = 3        

M∩S = 5 (but 3 are already accounted for) so 2 for M and S ONLY

M∩E = 7 (but 3 are already accounted for) so 4 for M and E ONLY

S∩E = 8 (but 3 are already accounted for) so 5 for S and E ONLY

M = 14 (but 3 + 2 + 4 are already accounted for) so 5 for M ONLY

S = 20 (but 3 + 2 + 5 are already accounted for) so 10 for S ONLY

E = 28 (but 3 + 4 + 5 are already accounted for) so 16 for E ONLY

Therefore, the students already accounted for is 3 + 2 +4 + 5 + 5 + 10 + 16 = 45 students

So, those students who don’t like any of these subjects are 50 – 45 = 5 students

A = {2, 4, 6, 8, 10, 12}

B = {3, 6, 9, 12, 15, 18}

The intersection of a set means that the elements are in both sets: A∩B = {6, 12}

This problem can be solved two ways, with a formula or with reason.

Using the formula, the intersection of the Venn diagram for which classes students take is:

By using reason, it is clear that 60 + 70 is greater than 100 by 30. It is assumed that this extra 30 students come from students who were counted twice because they took both classes.