Video: Using Venn Diagrams to Decide Whether Certain Conclusions Are Valid

Consider the following Venn diagram. Which conditional statement is represented by the diagram? [A] If a student scored over 90%, then they got an A [B] If a student got an A, then they scored over 90%. Use the Venn diagram to decide whether the following statement is valid: If Benjamin scored less than 90%, then he did not get an A.

02:39

Video Transcript

Consider the following Venn diagram. In the larger circle on the outside are students who got an A. And a subset of that, inside of that, students who scored over 90 percent. Which conditional statement is represented by the graph? a) If a student scored over 90 percent, then they got an A. Or b) If a student got an A, then they scored over 90 percent.

To help us think rightly about this, let’s look at three students: a student who’s in the smallest circle, a student who got an A but is not in the over 90 percent circle, and we have a third student on the outside who did not get an A. Both of these students got an A. But only one of them scored over 90 percent.

We do have a student who got an A but did not score over 90 percent. And that means statement b is impossible. Because it says: If a student got an A, then they scored over 90 percent. And we can produce a counterexample to that statement. However, if a student scored 90 percent, they got an A. Because students who scored over 90 percent is a subset of students who got an A, all students who score over 90 percent will get an A.

Now, we need to find out if the following statement is valid: If Benjamin scored less than 90 percent, then he did not get an A.

One thing we could try to do is place Benjamin in one of these categories. Benjamin scored less than 90 percent. We have a student outside who scored less than 90 percent. However, some of the students that did not score over 90 percent were still given an A. This statement says he did not get an A.

While we cannot say for certain that Benjamin got an A, we can’t rule it out. We can’t say that he didn’t. It is possible that he got an A. The statement “If Benjamin scored less than 90 percent, then he did not get an A” is not valid.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.