Practice problems: (Section 1.2) 1-43, odd.

Assigned problems:
1. Consider the following two properties: (a) A∈B, and (b) A⊆B.
(i) Give examples of sets A and B where property (a) holds but (b) fails.
(ii) Give examples of sets A and B where property (a) fails but (b) holds.
(iii) Give examples of sets A and B where properties (a) and (b) both hold.
(iv) Give examples of sets A and B where properties (a) and (b) both fail.

2. If A and B are sets, define (A,B) to be the set {{A},{A,B}}. (A,B) is called an ordered pair. Using the axioms of set theory, explain why
(i) (A,B) is a legitimate set. (That is, show that the axioms of set theory guarantee that if A and B are sets, then (A,B) is also a set.)
(ii) (A,B) = (C,D) if and only if A = C and B = D.

3. Prove that 1·1=1.

Practice problems: 1-39, odd, except for problems involving products or powers of sets.

Assigned problems:
1. Prove or disprove:
(i) P(A∩B) = P(A)∩P(B).
(ii) P(A∪B) = P(A)∪P(B).
(Here P = power set.)

2. Prove or disprove: for any A, B and X, if A∪X = B∪X and A∩X = B∩X, then A=B.

3. Show that A⊆B if and only if A∩B=A.

Solutions.

Practice problems: (Section 1.1) 5-9, 17-53, odd, except for problems involving ⊕.

Assigned problems:
1. Verify the following logical equivalences.
(i) ((¬p)→(¬q))≡(q→p)
(ii) (p→(q→r))≡((p∧q)→r))
(iii) ((p∧q)→r)≡((p∧(¬r))→(¬q))

2. Write the following propositions in disjunctive normal form, assuming that each proposition is a function of p, q and r.
(i) (p→r)
(ii) ((p↔q)→((¬p)↔r))
(iii) q

3. Draw formula trees for the following propositions.
(i) ((p∧(¬r))→(¬q))
(ii) (¬ ((¬(p∨q))∧ (¬(¬((¬p)→r)))))

2. In 1959, Pete Seeger took lines from the Book of Ecclesiastes to write a song, which was made famous by the Byrds in 1965. One line is:

To every thing there is a season, and a time for every purpose under heaven.

Write this as a formal sentence using predicates S(s,t) = "s is the season for thing t" and P(T,p) = "T is the time for purpose p".

3. Write a formula Prime(x) that is true for a natural number x if it is prime and is false for all other natural numbers. Use whatever standard symbols you need.

2.
(i) Write ((∀ x P(x))→(∃ x P(x))) in prenex form.
(ii) Use a quantifier game to prove that the sentence in (i) is true in any nonempty structure that has a predicate symbol P.

3. Let G(x) =
  • 1/x if x is not zero
  • 0 if x = 0
Use quantifier games to show that G is not continuous at x=0, but is continuous at x=1.

Assigned problems:
1. (Restricted quantifiers)
(i) Using the definition of restricted quantifiers and some logical equivalences that you know show that
¬((∃x P(x)) Q(x))
is logically equivalent to
(∀x P(x)) (¬Q(x)).
(ii) Write
(∀ x>0)(∃ y≠ x)(∀ z≠ y)(x<z)
as an equivalent sentence that does not use restricted quantifiers.

2. Explain why the following are valid sentences. (Here P(x) has x as its only free variable and Q and R have no free variables.)
(i) (∀ x)((¬ P(x))∨ P(x))
(ii) (∀ x)(P(x) → P(x))
(iii) (Q→(R→Q))

3. A hypothetical syllogism is a deduction of the following form: from P→Q and Q→R, infer P→ R. Explain why
{P→Q,Q→R}|=P→ R.

Practice problems: (Chapter 2 Review) 2, 4, 5, 8, 9, 19, 25, 27, 28, 34, 36, 37, 38, 39, 40.

Assigned problems:
1. Show that if A⊆B and C⊆D, then A∪C⊆B∪D. (Clearly identify separate cases when case division is used.)

2. Show that "A⊆B and B⊆A implies A=B" in each of the following two ways.
(i) With a direct proof.
(ii) With a proof by contradiction.

3. Show that "any nonconstant, real, linear function f(x)=ax+b has a unique root" in each of the following two ways.
(i) With a direct proof.
(ii) With a proof by contradiction.

Practice problems: (Chapter 2 Review) 1-11 odd, 17, 21, 33, 35

Assigned problems:
1.
(i) Exercise 4.3 (10).
(ii) Exercise 4.3 (12).

2. Exercise 4.4 (36).

3. You want to divide a piece of chocolate into 1"x1" square pieces to distribute to a group of children. The starting piece is an m"x n" rectangle. Show that it can be divided by breaking pieces of chocolate a total of mn-1 times, but not with fewer breaks. (Each "break" divides only one piece, and must be a straight line parallel to one of the sides of the piece. At each stage, all pieces should be rectangles whose sides have integer length.)

Assigned problems:
1.
(i) Exercise 5.3 (15).
(ii) Exercise 5.3 (16).
(iii) Exercise 5.3 (40).

2.
(i) Exercise 5.4 (14).
(ii) Exercise 5.4 (40).
(iii) Exercise 5.4 (42).

3.
(i) Chap 5 Review (6).
(ii) Chap 5 Review (8).
(iii) Chap 5 Review (43).

Assigned problems:
1. Describe a binary relation on some set that is:
(i) reflexive and symmetric but not transitive.
(ii) reflexive and transitive but not symmetric.
(iii) symmetric and transitive but not reflexive.

2. Prove or disprove:
(i) The union of two equivalence relations is an equivalence relation.
(ii) The intersection of two equivalence relations is an equivalence relation.

3.
(i) Exercise 5.2 (40).
(ii) Exercise 5.2 (56).
(iii) Exercise 5.2 (60).

Assigned problems:
1. Let f:A→B and g:B→C be arbitrary functions. Prove that ker(f)⊆ker(gοf).

2. Let E be an equivalence relation on a set X. Under what circumstances is the function f: X/E→X: [x]_E↦x well defined?

3.
(i) Exercise 3.6 (18).
(ii) Exercise 3.6 (28).

Assigned problems:
1. Exercise 5.6 (20).

2. Section 6.1
(i) Exercise (14).
(ii) Exercise (18).
(iii) Exercise (28).

3. Section 6.2
(i) Exercise (10).
(ii) Exercise (20).
(iii) Exercise (22).