Homework 2
due: February, 7, 2013
Problems marked with ∗ will be graded. Please practice on the rest of them to the extent you find useful.
1. Rewrite the following sentences, using quantifiers (∃, ∀) and logical operators(¬, ∧, ∨, ⇒, ⇔), where
possible.
(a) ∗There is someone older that 21
(b) Everyone in your class is friendly
(c) Every student must take at least 60 course hours, or at least 45 course hours and write master
thesis, and receive a grade no lower than B in all required courses, to receive a master’s degree.
(d) ∗For all natural numbers n, there exist a natural number m, such that n + 5 = m
2. ∗Rewrite the sentence below, using ¬∃x p(x) = ∀x ¬p(x), as well as ¬∀x p(x) = ∃x ¬p(x)
¬ (∃N >0 ∀n>N
1
1
<
)
n 100
3. Let A = {1, 2, 3, 4}, B = {2, 5, 6}, C = {3, 5, 6}, calculate:
(a) ∗A ∩ B
(b) A ∖ B
(c) (A ∪ B) ∖ C
(d) ∗(A ∖ B) ∪ (A ∖ C) ∪ (C ∖ A)
4. Find examples of three sets A, B, C such that
(a) A ≠ B and A ∪ C = B ∪ C
(b) ∗A ≠ B and A ∩ C = B ∩ C
5. [(p ⇒ q) ∧ (q ⇒ r)] ⇒ (p ⇒ r) — prove it using table of equivalences(see HW1). Note that this
sentence tells us a very basic law of deduction.
6. ∗[(A ⊂ B) ∧ (B ⊂ C)] ⇒ (A ⊂ C) — prove it. Hint: use the proposition from the previous problem.
7. Prove the following identities using notation {x ∶ something} = {x ∶ something else} = ..., and the
table of equivalences (see HW1).
(a) A ∪ (B ∪ C) = (A ∪ B) ∪ C
(b) ∗A ∩ (B ∩ C) = (A ∩ B) ∩ C
(c) A ∩ A = ∅
8. Write a proof, that uses words: ”Assume, that something, so something else, thus we can conclude
that something...”
(a) A = B ⇐⇒ (A ⊂ B ∧ B ⊂ A)
(b) ∗A ∩ B ⊂ A
(c) A ⊂ B ⇐⇒ B ⊂ A
9. Draw a Venn diagram for:
(a) A ∖ (B ∩ C)
(b) ∗A ∪ (B ∖ C)
(c) (A ∪ B) ∖ C
(d) A ∩ (B ∩ C)
1
10. If true, write a proof using the table of set identities(see below), if false find a counterexample
(find sets for which it is not satisfied). Hint: remember, that the Venn diagram is a useful tool in
determining whether a set identity is true or false.
(a) ∗(A ∖ B) ∪ (A ∖ C) = A ∖ (B ∩ C)
(b) A ∩ B ⊂ A ∪ B
(c) (A ∩ B) ∖ C = B ∩ C
(d) ∗A ∩ B ∩ C = B ∖ (A ∩ B)
11. ∗Prove identity 3) using other identities from the table.
12. Prove identity 23) using other identities from the table.
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
18)
19)
20)
21)
22)
23)
24)
25)
Table of set identities
A∩U =A
A∪∅=A
A∪U =U
A∩∅=∅
A∪A=A
A∩A=A
(A) = A
A∪B =B∪A
A∩B =B∩A
A ∩ (B ∩ C) = (A ∩ B) ∩ C
A ∪ (B ∪ C) = (A ∪ B) ∪ C
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
A∩B =A∪B
A∪B =A∩B
A ∪ (A ∩ B) = A
A ∩ (A ∪ B) = A
A∪A=U
A∩A=∅
A = B ⇐⇒ (A ⊂ B ∧ B ⊂ A)
A∖B =A∩B
A∩B ⊂A
A⊂A∪B
(A ⊂ B) ∧ (B ⊂ C) ⇒ (A ⊂ C)
A ⊂ B ⇐⇒ B ⊂ A
2