Sets

Questions?

Introduction to Sets

Parts of section 2.3 of our textbook.

Roster Notation

Use roster notation to write the set of majors represented in our class: { Education, Math, Physics }. This is the simplest form, a short list of elements inside braces. Notice that the elements don’t have to be numbers or anything particularly mathematical.

Use roster notation to write the following:

• A: the set of perfect squares that are between but not including 0 and 10. { 1, 4, 9 }. Another simple case.
• B: the set of integer multiples of 3 that are between 1 and 100. { 3, 6, 9, ..., 99 }. This uses ellipsis to avoid having to write a repeating pattern more than necessary.
• C: the set of real numbers that are the positive square root of some integer. { √1,√2, √3, ... }. Ellipsis indicating a pattern that repeats forever. This could also be { 1, √2, √3, 2, ... } but consistently using the radical sign makes the pattern clearer.
• ℕ: the set of natural numbers. { 1,2,3, ... }. Ellipsis to indicate an infinitely continuing pattern again.
• D: The set of all integer multiples of 3. { ..., -9, -6, -3, 0, 3, 6, 9, ... }. Ellipsis to indicate a pattern that repeats forever both before and after the part of the set shown explicitly.
• E: The set of pink elephants dancing on tables in our classroom. ∅ (empty set).

Set Concepts and More Notation

Are any of the above sets subsets of others? Which ones? List the subset relationships using the ⊆ symbol.

B ⊆ D, A ⊆ ℕ, ℕ⊆C, verified by checking that every element the first set in each example is a member of the second.

E is a subset of all of them, i.e., the empty set is a subset of everything.

Every set is a subset of itself, too.

For which of the above sets, X, is it true that 0 ∈ X? D, for example.

Are any of the above sets equal to each other? No, none contain exactly the same elements.

Proofs about Sets

Prove the following theorem: The set of real-valued solutions to the equation x² - 4 = 0 is a subset of the set of real-valued solutions to the equation x³ + x² - 4x - 4 = 0.

The key goal is to show that every solution to x² - 4 = 0 is also a solution to x³ + x² - 4x - 4 = 0. There are several ways you could do this, but the most elegant we heard uses factoring. Formally, a proof might go like this (notice that I used the definition of subset to restate the theorem):

Theorem: If y is a real-valued solution to x² - 4 = 0, then y is also a solution to x³ + x² - 4x - 4 = 0.

Proof. We assume that y is a solution to x² - 4 = 0, and will show that y is also a solution to x³ + x² - 4x - 4 = 0. Since y is a solution to x² - 4 = 0, we have that y² - 4 = 0. Now observe that x³ + x² - 4x - 4 factors into

x³ + x² - 4x - 4 = (x² - 4)(x+1).

Plugging y into this equation yields

y³ + y² - 4y - 4 = (y² - 4)(y+1)

= 0

where the last equality holds because y² - 4 = 0. We have thus shown that if y is a real-valued solution to x² - 4 = 0, then y is also a solution to x³ + x² - 4x - 4 = 0. QED.

So now you’ve seen how to build proofs that one set is a subset of another around the definition of “subset”: to show the X is a subset of Y you just have to show that every member of X is also a member of Y, which in turn is just the conditional “if x is a member of X, then x is a member of Y.”

Next

Roster notation is limited in what it can describe.

Predicates and set builder notation.

Read the rest of section 2.3 of the textbook (preview activity 2, “Variables and Open Sentences,” “Set Builder Notation,” “When the Truth Set is the Universal Set”).