SUNY Geneseo Department of Mathematics
Friday, February 2
Math 239 01
Spring 2018
Prof. Doug Baldwin
Parts of section 2.3
Some sets about us (e.g., favorite colors, majors, kinds of pet, favorite sports, etc.)
Majors = { chem, phys, bio, math, edu, sped }. Note the basic notation for a small, finite set. Also note that even if there are several students with the same major, it only appears once -- sets don’t contain duplicates.
Your favorite subjects = { math, phys, chem }. This is a subset of the majors and of our favorite subjects. Everything in this set is also in those other sets.
Our favorite subjects = { math, phys, chem, cs, hist }
Made up favorite subjects = { math, phys, chem, edu, sped, bio }. This set is made up specifically to illustrate two equal sets -- it equals the set of majors, despite being in a different order. Order doesn’t matter in sets.
Write the set of integer multiples of 3 between 3 and 18. { 3, 6, 9, 12, 15, 18 }. This is another use of the simplest roster notation.
How would you describe { 3, 6, 9, 12, ..., 300 } in an English sentence? Integer multiples of 3 between 3 and 300. When a large set has elements that follow a recognizable pattern, you can list just enough elements to establish the pattern and show where it ends. The ellipsis (“...”) indicates that there are other elements from the pattern that aren’t shown explicitly.
Write the set of all positive integer multiples of 3. { 3, 6, 9, ... }. Ellipsis can appear at the end or beginning of a list of elements to show that a pattern continues infinitely. As a preview of upcoming notation, this could also be written as { 3n | n ∈ ℕ }.
How about all integer multiples of 3? { ..., -6, -3, 0, 3, 6, ... }. Note the ellipsis at both ends of a list.
Write the set of all integer multiples of 6. { ..., -12, -6, 0, 6, 12, ... }.
What is the set of all integer multiples of 6 that are odd? This is an example of the empty set, written {} or ∅.
Is the set of integer multiples of 6 a subset of the set of integer multiples of 3? Vice versa? Can you prove either claim?
The multiples of 6 are a subset of multiples of 3. And it can be formally proven...
Theorem: The set of integer multiples of 6 is a subset of the set of integer multiples of 3.
Proof: We will show that the set of integer multiples of 6 is a subset of the set of integer multiples of 3 by showing that every integer multiple of 6 is an integer multiple of 3. Assume that x is an integer multiple of 6, i.e., x = 6n for some integer n. Factoring 6 into 3 × 2 and the regrouping, we have
x = (3 × 2)n
= 3 × (2n)
Since the integers are closed under multiplication, 2n is an integer, and so in turn 3 × (2n) is an integer multiple of 3. We have thus shown that every integer multiple of 6 is an integer multiple of 3, and thus that the set of integer multiples of 6 is a subset of the set of integer multiples of 3. QED.
The multiples of 3 are not a subset of the multiples of 6.
Counter-example: 3 is a multiple of 3 but not of 6.
The definition of “set” and the properties that make a group of things a set.
Roster notation for defining sets.
The empty set.
How to prove that one set is a subset of another.
Using counter-examples to show that a claim is not true.
Variables and predicates.
Read the parts of section 2.3 that you didn’t read for today, i.e.: