Lesson Seven - Review of Set Theory

Set theory uses the Venn Diagram to visually express the relationship between sets and among elements.

Example 1

For the questions below, refer to this set.
A = {cheese, 1, unicorn, cat, donkey}
B = {headphones, skittles, cheese, 1}

Can you draw a Venn Diagram with sets A and B including the elements?

  1. What is the intersection of A and B {A  B}?
  2. What is the union of A and B {A  B}?
  3. What is the set difference of A and B {A \ B}?
  4. What is the multiplication of A and B {A • B}?

Laws of Set Theory

Remember that the Cardinal Number of a set is the total number of elements in that set. What are the cardinal numbers for set A and Set B above? Count the number of elements in each. Set A has [cheese, 1, unicorn, cat, donkey]. This makes 5. Set B has [headphones, skittles, cheese, 1]. This makes four. Answer = 5 and 4.

We can write this as an absolute value of Set A and Set B.
|A| = 5, |B| = 4

Using these absolute values, you can also make some laws that help calculate.

  1. |A  B| ≤ |A|, The intersection of Sets A and B is greater than or equal to the absolute value of Set A*

  2. |A  B| ≤ |B|, The intersection of Sets A and B is greater than or equal to the absolute value of Set B*

  3. *In other words, the combination of Sets A and B must be larger or equal to one of the included Sets. It cannot be smaller.

  4. |A  B| ≤ |A| + |B|
    If |A  B| = |A| + |B|, then A  B = { }
  5. |A \ B| ≤ |A|
  6. |A x B| = |A| • |B|

A  B, A is not a subset of B
B  A, B is not a subset of A
D  A, D is a subset of A
D  A  |D| ≤ |A|, If D has more elements than Set A, then D cannot be a subset of A.

Set theory can be very helpful in determining reasoning errors. A violation of set theory is a reasoning error, or fallacy.

Read the following sentence, and decide whether you agree or disagree:
“Cutting people is a crime.”

What do you think? “Cutting people a crime” is true, right? Are you sure? Not exactly. This statement illustrates the first type of fallacy that we are going to cover: fallacy of accident or sweeping generalization.

The fallacy of sweeping generalization occurs when a generalization ignores exceptions.

Set theory can help us in evaluating statements, so let’s use set theory to see if we can come up with any instances in which cutting people is not a crime.

This example illustrates the fallacy of accident or sweeping generalization. This means that it disregards exceptions when making an argument. Ignoring an exception is poor logic.

Lesson Seven Exercise

Who at the candy?i Venn diagrams are used for a variety of games. Take the following riddle: Jen, Maggie, and Nicole often eat lunch together, but we don’t know which one eats candy after lunch. However, we know that

  1. If Jen gets candy, so does Maggie.
  2. Maggie or Nicole will sometimes get candy, but they never get candy at the same time.
  3. Jen or Nicole sometimes get candy, either one of them or both of them.
  4. If Nicole gets candy, so will Jen.

Let J = Jen eats the candy; M = Maggie eats the candy; N = Nicole eats the candy.