Homework #2
CSI2101: Discrete Mathematics (Fall 2023)
Instructions
-
Solve the following problems by writing down hand-written answers on papers, tablets, etc.
Scan/convert your hand-written answers into a single PDF file
Upload the PDF file to LearnUs by 11:59 pm KST on October 22, 2023 (Sun)
Problems
|𝐴|
Q1: Show that if 𝐴 is a finite set with |𝐴| = 𝑛, then |𝑃(𝐴)| = 2 . Note that 𝑃(𝐴) refers to the power set of 𝐴.
Q2: Suppose we do not accept ordered pairs as primitive, and instead define 〈𝑥, 𝑦〉 to be {𝑥, {𝑥, 𝑦}}. Prove
that 〈𝑥, 𝑦〉 = 〈𝑢, 𝑣〉 if and only if 𝑥 = 𝑢 and 𝑦 = 𝑣.
−1
Q3: Let 𝑓 be any function. Suppose that the inverse relation 𝑓
bijection? Explain.
−1
= {〈𝑦, 𝑥〉: 𝑦 = 𝑓(𝑥)} is a function. Is 𝑓
−1
Q4: Suppose 𝑓: 𝐴 → 𝐵, 𝑔: 𝐶 → 𝐷, and 𝐴 ⊆ 𝐷. Explain when (𝑓 ◦ 𝑔)
−1
𝐵 to 𝐶, and express it in terms of 𝑓
a
exists as a function from a subset of
−1
and 𝑔 . Note that (𝑓 ◦ 𝑔)(𝑥) = 𝑓(𝑔(𝑥)).
Q5: Two sets have the same size if there is a bijection between them. Prove that there are infinitely many
different sizes of infinite sets; that is, that there are at least a countably infinite number of infinite sets, no
two of which have the same size.
Q6: Prove that the union of countably many countably infinite sets is countably infinite.
Q7: If 𝑎 is a symbol and 𝑠 is a string, define #𝑎(𝑠) to be the number of occurrences of 𝑎 in 𝑠. For example,
#1(101100101) = 5. Define #𝑎(𝑠) inductively, and then prove by structural induction that for any strings 𝑠
and 𝑡, #𝑎(𝑠 · 𝑡) = #𝑎(𝑠) + #𝑎(𝑡).
Q8: A subsequence of a string 𝑠 is a sequence of the symbols from 𝑠 in the order in which they appear in 𝑠,
though not necessarily contiguously. If we write 𝑆𝑆(𝑠) for the set of all subsequences of 𝑆, then, for
example, 𝑆𝑆(101) = {λ, 0, 1, 10, 01, 11, 101}. Define 𝑆𝑆(𝑠) inductively.