The symbol \(\displaystyle \cup\) stands for the union between two sets.  Therefore, \(\displaystyle A\cup B\) means the set of all numbers that are in either A or B.  Looking at our choices, the only number that isn't in either A, B, or both is 23.

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.

\(\displaystyle 50+29=79\)

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 \(\displaystyle \cup\) stands for "union," which refers to everything that is in either set. \(\displaystyle \textup{Calculus} \cup\textup{Spanish}\) 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.

\(\displaystyle 24+ 29 = 53 > 40\)

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.

\(\displaystyle 53 - 40 = 13\)

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 \(\displaystyle A \cup C\) stands for "A union C," which refers to everything that is in either set \(\displaystyle A\) or set \(\displaystyle C\).

\(\displaystyle A\cup C= \left \{ 12, 2, 3, 4, 11, 5 \right \}\)

When we add the numbers together, we get:

\(\displaystyle 12 + 2 + 3 + 4 + 11 + 5 = 37\)

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.

\(\displaystyle 15 - 5 = 10\)

Ten students have only cats.

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

\(\displaystyle 12 - 5 = 7\)

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.

\(\displaystyle 10 + 7 + 5 + 8 = 30\)

\(\displaystyle A\cap B\) means the intersection of the sets \(\displaystyle A\) and \(\displaystyle B\), 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: \(\displaystyle \null\)\(\displaystyle \emptyset\)

\(\displaystyle A \cup B\) is the union of the sets \(\displaystyle A\) and \(\displaystyle B\) which is the set that contains anything that is in either set. Thus, the total of \(\displaystyle A \cup B\) is every element that is in one of the two sets, and so \(\displaystyle A \cup B\) \(\displaystyle =\{1,2,3,4,5,6,7\}\)