Homework #4: Sets and functions
MATH 174
1. Let U be the set of all MATH 174 students. Let A be the subset of U consisting of CS
majors or minors; let B be the subset of U consisting of IS majors or minors; and let C be
the subset of U consisting of math majors or minors.
(a) Describe A ∩ B in words and draw a Venn diagram for this set.
(b) Describe C in words and draw a Venn diagram for this set.
(c) Describe A ∪ B ∪ C in words and draw a Venn diagram for this set.
(d) Describe (C − B) ∪ (A − B) in words and draw a Venn diagram for this set.
(e) Using A, B, C, and the set operations, build a set of which you are a member.1
2. Let X = {H, T }, where H and T are just symbols.
(a) Write P(X) (as a roster).
(b) Let p : P(X) → [0, 1] be the function defined in the following way. Let E ∈ P(X).
If |E| = 0, then p(E) = 0.
If |E| = 1, then p(E) = 0.5.
If |E| = 2, then p(E) = 1.
Compute P (E) for all E ∈ P(X).
(c) The function p appears very naturally in our world. Where does p show up in “real
life”? (Hint: What do H and T stand for?) (Full credit awarded for a good guess.)
3. Determine whether the following functions are injective, surjective, bijective, or none of
the above. If bijective, find the function’s inverse.
(a) Let f be the function that takes a CCU student and returns that student’s CINO number.
(b) Let g : N → N be defined by g(n) = n2 .
(c) Let h : R → R be defined by h(x) = x2 .
(d) Let F : R → R≥0 be defined by F (x) = x2 . (The symbol R≥0 denotes the non-negative
real numbers.)
4. Prove one of the following.
(a) Let A and B be sets. Prove DeMorgan’s laws: A ∪ B = A ∩ B and A ∩ B = A ∪ B.
(b) Suppose f : R → R is strictly increasing, i.e. that for all x, y ∈ R, if x < y, then
f (x) < f (y). Prove f is injective.
1 One
student’s answer will be A ∪ B ∪ C!
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