Mathematics 220 Workshop 1 - Set Theory
Note: Your goal is to practice working with notation and definitions. Some problems will
require knowledge of definitions - as you encounter a definition that you have to look up, then
do so and immediately learn it by heart.
1. Which of the following sets are subsets of others on the list? Consider all 12 possibilities.
• A = {n ∈ N : n is odd}
• B = {n ∈ N : n is prime}
• C = {4n + 1 : n ∈ N}
• D = {x ∈ R : x2 − 39x + 74 = 0}
2. Let
A = {n ∈ N : n = 4k for some k ∈ N, 1 ≤ k ≤ 10},
B = {n ∈ N : n = 3k for some k ∈ N, 1 ≤ k ≤ 30}.
Calculate the following:
(a) |A| and |B|
(b) |A ∩ B|
(c) |A ∪ B|
(d) Put together your thought process for items (c), and (d) and guess a formula for
|A ∪ B| in terms of |A|, |B|, and |A ∩ B|. Draw Venn Diagrams to help you.
(e) Guess a similar formula for three sets.
3. Let A and B be subsets of a unversal set U. Prove that A ∪ B = A ∩ B.
Do this by showing that each side is a subset of the other side. A Venn diagram may help
convince you this is true but is not a proof.
Sets in geometry - practice using a picture or a graph to help you solve the problem.
However, keep in mind: a picture is not a solution! Mainly, it is a guide for yourself to
obtain intuition about the solution
4. Let C be the circle of radius 1 centered at the origin in the xy-plane. Express this set in
terms of its points (x, y), and some property p(x, y).
5. Let a, b, c, d be real numbers with a < b < c < d. Express the set [a, b] ∪ [c, d] as the
difference of two sets.
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Mathematics 220 Workshop 1 - Set Theory
Power sets
6. Compute the power set P ({{1}, 1, 2})
7. Binomial coefficients and the power set. Let A be a set of cardinality n. In this
question we get to see how the number of k-element subsets of A adds up to 2n as k runs
through the numbers from 0 to n.
(a) Warm-up: let A = {1, 2, 3, 4, 5}. How many 3-element subsets does it have?
(b) Consider the expressions
(a + b)2 = a2 + 2ab + b2
(a + b)3 = a3 + 3a2 b + 3ab2 + b3
...
(a + b)n = an + C1 an−1 b + C2 an−2 b2 + · · · + Cn−1 abn−1 + bn .
We are interested in these coefficients C1 , . . . Cn−1 (we can also write C0 = Cn = 1),
called binomial coefficients. We do not yet know a formula for these coefficients,
but we already can prove something about them!
Prove that C0 + C1 + · · · + Cn = 2n .
Hint: consider the expression (1 + 1)n .
(c) Prove that for every k = 0, 1, ...n, the coefficient Ck equals the number of k-element
subsets of A.
Hint: First consider small n and small k, and convince yourself that this is true.
Then consider the expression
(a + b)n = (a + b)(a + b)(a + b)....(a + b).
Let us number these factors: the first (a + b), the second (a + b), and so on, all
the way to the n-th (a + b). Now, when we are multiplying it all out, from each
(a + b) we get to pick either a or b. If we get the product ab bn−k as a result of this
picking of as and bs, then it means that we picked a precisely k times, and we picked
b the rest of the time. Thus, the coefficient Ck is the number of ways to select k
a’s. Let us mark the factors from which we picked a. We get a k-element subset
of A = {1, . . . , n}!. Thus we have matched up the ways to pick k factors a, and
k-element subsets of A. Then there are precisely Ck subsets of k elements.
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