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CHAPTER 1
Logic 1
Definitions
D1. Statements (propositions), compound statements.
D2. Truth values for compound statements p ∧ q, p ∨ q, p → q, p ↔ q. Truth
tables.
D3. Converse and contrapositive.
D4. Tautologies and contradictions. Logical equivalence.
D5. Propositional forms (predicates). Quantifiers ∀ and ∃.
Facts (with proofs)
F1.
F2.
F3.
F4.
F5.
An implication and its contrapositive are logically equivalent statements.
(p ↔ q) ⇔ (p → q) ∧ (q → p)
De Morgan’s Laws.
Distributive Laws for conjunction and disjunction.
The negation of a statement involving one quantifier: ¬(∀xP (x)) ⇔
∃x(¬P (x)) and ¬(∃xP (x)) ⇔ ∀x(¬P (x)). Negation of the statements
involving several quantifiers.
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1. LOGIC 1
Problems
Easier
1.1 Let p and q and r be the propositions:
p = “m/3 is an integer”
q = “n/2 is an integer”
r = “(nm)/6 is an integer”
1.2
1.3
1.4
1.5
1.6
1.7
1.8
Write the following statements using p, q, r and the symbols ∨, ∧, ¬, →
and ↔.
(a) “If m/3 is an integer and n/2 is an integer, then (nm)/6 is an integer”.
(b) The converse of your statement from part (a) above.
(c) The contrapositive of your statement from part (a) above.
(d) Is the statement in part (b) above true or false? Give your reasons.
Give truth tables for the following statements. Determine which are tautologies, which are contradictions and which are neither.
(a) ¬[(p ∧ ¬p) → q].
(b) ¬(q → p) → ¬p.
(c) (p ∧ q) ∧ ¬(p ∨ q).
(d) (p → q) → r.
(e) [(p → q) ∧ (q → r)] → ¬(p → r).
Suppose that the truth values of the statements p and r are true and the
truth values of the statements q and s are false. Find the truth values of
the following compound statements.
(a) ¬(¬(¬p ∨ q) ∨ r) ∨ s.
(b) (p → q) → (r → s).
Prove De Morgan’s Laws:
(a) ¬(p ∨ q) ⇔ ¬p ∧ ¬q.
(b) ¬(p ∧ q) ⇔ ¬p ∨ ¬q.
Prove that ¬(p ↔ q) is logically equivalent to ¬(p → q) ∨ ¬(q → p).
Let p and q be compound statements. If p is a contradiction, what can be
said about the truth value of statement p → q? If p is a tautology, what
can be said about the truth value of statement p → q? If q is a tautology,
what can be said about the truth value of the statement p → q?
Find the truth values of the following statements. Then construct their
negations.
(a) There exists an integer x such that (x2 − x − 1)(x + 1) = 0.
(b) For all real numbers x and y, x2 + y 3 ≥ 0.
Find the truth values of the following statements.
(a) For all real x, (x2 − x − 1)(x + 1) = 0 implies x2 − x − 1 = 0.
(b) For all real numbers x, (x2 +1)(x3 −x−10) = 0 implies x3 −x−10 = 0.
2
2
(c) For all real numbers x, x
√ (x + 1) > 1 implies x(x + 1) > 1.
(d) For all real numbers x, 5x + 1 < 9 implies x < 16.
1. LOGIC 1
3
Medium
1.9 Is the sentence “This statement is false” a proposition?
1.10 Determine how many rows a truth table of a compound statement would
have if that involved 2,3,4 distinct simple statements, respectively. As
examples, consider compound statements ¬p ∧ ¬q, p ∨ q ∨ ¬r, (p ∨ q) →
(¬(r∧t))? You do not have to construct the complete tables to answer this
question. Do you see any pattern in your answers? What is the answer if
a compound statement is constructed out of n simple statements?
1.11 Let P (x) and Q(x) be two predicates over reals (i.e., x is a real number).
Show that
(∃x such that P (x)) ∧ (∃x such that Q(x))
and
1.12
1.13
1.14
1.15
1.16
1.17
∃x such that (P (x)) ∧ Q(x)
are not necessarily logically equivalent.
Hint: It is sufficient to find examples of such P (x) and Q(x).
Let P (x, y) be the predicate x = 2y + 1, where x and y are integers. Find
the truth values of the following statement and explain your answer.
(a) ∃x∃y P (x, y)
(b) ∃x∀y P (x, y)
(c) ∀x∃y P (x, y)
(d) ∀x∀y P (x, y)
Find a compound statement A made out of propositions p, q and r with
the property that
(a) A is True if all p, q, r are True, and A is False in all other cases.
(b) A is True if all p, q, r are True, A is True if all p, q, r are False, and
A is False in all other cases.
Determine whether the following statements are true or false. Assume
that x, y are reals. Prove your answers.
2
2
≥ xy).
(a) ∀x∀y ( x +y
2
(b) ∃x∃y ( xy + xy < 2).
(c) ∃x > 0 ∃y > 0 ( xy + xy < 2).
Find the truth values of the following statements.
(a) For all real numbers x, x2 − 7x ≥ 12.
(b) For all real numbers x and y, x2 − 3xy + y 2 ≥ 0.
(c) For all real numbers x and y, x2 + xy + y 2 ≥ 0.
Let a, b, c be arbitrary real numbers.
(i) Show that if a + b + c = 0, then a3 + b3 + c3 = 3abc.
(ii) State the converse statement to (i). Is it true?
Suppose a, b, c are integers. Are the following statements true? Prove
your answers.
(a) ∀a∃b∃c (b2 + c2 = a2 ).
(b) ∀a 6= 0 ∃b 6= 0 ∃c 6= 0 (b2 + c2 = a2 ).
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1. LOGIC 1
Harder
1.18 Is the following statement true?
For every integer n, there exist integers x and y such that
x2 − y 2 = 2n + 1.
Prove your answer.
1.19 Prove that for all real numbers x, y, z,
x2 + y 2 + z 2 ≥ xy + xz + yz.
Moreover, prove that the equality sign is achieved if and only if x = y = z.
1.20 Prove that for all real numbers x, y, z,
1 1
1
1
1
1
1
1
+ + =
implies 5 + 5 + 5 =
.
x y z
x+y+z
x
y
z
(x + y + z)5
1.21 Suppose a3 + b3 + c3 = 3abc, and numbers a, b, c are not all equal. Does
it imply imply a + b + c = 0?
1. LOGIC 1
5
Some Answers and Hints
Note that most of the comments below are hints. More details are needed when
you explain/proof your claims.
Easier
1.1
1.3
1.4
1.5
1.6
1.7
1.8
(1d) False.
(3a) False ; (3b) True
Use truth tables.
Use truth tables.
True; can be either True of False; True
(7a) True; (7b) False.
(8a) False; (8b) True; (8c) True; (8d) True.
Medium
1.9 No. If it is true, then it is false. If it is false, so it is true. This is an
example of how a self-reference may create a problem.
1.10 4; 8; 16; 2n .
1.12 (12a) True; (12b) False; (12c) False; (12d) False.
1.13 (13a) Easy; (13b) Can use your answer to (13a).
1.14 (14a) True; (14b) True; (14c) False.
1.15 (15a) False. Find a counterexample. (15b) False. Find a counterexample.
(15c) True. Complete the square.
1.16 Rewrite the first condition as a = −b − c.
1.17 (17a) True. Remember that 0 is an integer. (17b) False. Find a counterexample.
Harder
1.18 Yes. If you have difficulty in finding a proof, let n be, e.g., 0, ±1, ±2, ±3, ±4, ±5.
In each of these cases find x and y as close to each other as possible. Try
to observe a pattern. Then generalize for an arbitrary integer n.
1.19 Use Problem 14a. (Make sure you know how to prove the statement you
are using).
1
imposes very strong restrictions on
1.20 The condition x1 + y1 + z1 = x+y+z
x, y, z. Try to understand what are they.
1.21 Yes it does. Hint: Factor a3 + b3 + c3 − 3abc.
CHAPTER 2
Set Theory
Definitions
D1. What do the notations N, Z, Q, R, C stand for?
D2. The empty set ∅, the power set P (A) of a set A, the Cartesian product
A × B of two sets A and B, the Cartesian n-th power An of a set A.
D3. Subsets, equality of sets, operations on sets (union, intersection, difference,
symmetric difference).
D4. Universal set, complement of a set.
Facts
F1. Important Laws:
(a) De Morgan’s Laws: (A ∪ B)c = Ac ∩ B c and (A ∩ B)c = Ac ∪ B c
(with proofs).
(b) Distributive Laws: A∩(B ∪C) = (A∩B)∪(A∩C) and A∪(B ∩C) =
(A ∪ B) ∩ (A ∪ C) (with proofs).
F2. If |A| = m, |B| = n, then |A × B| = mn.
F3. If |A| = n, then |P (A)| = 2n (with proof).
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2. SET THEORY
Problems
Easy
2.1 Let A = {1, 2, 3, {1}, {1, 3}}. Which of the following statements are true
or false. Explain your answers.
(a) 1 ∈ A; 1 ⊂ A; {1} ∈ A; {1} ⊂ A.
(b) 2 ∈ A; 2 ⊂ A; {2} ∈ A; {2} ⊂ A.
(c) {1, 2} ∈ A; {1, 2} ⊂ A.
(d) {1, 3} ∈ A; {1, 3} ⊂ A.
(e) {1, 2, 3} ∈ A; {1, 2, 3} ⊂ A.
(f) |A| = 5.
2.2 Let X1 = {M AT H210}, X2 = {M, A, T, H}, X3 = {210}, X4 = {M AT H},
and X5 = {2, 1, 0}. Find Xi ∩ Xj for i, j ∈ {1, 2, 3, 4, 5}.
2.3 Suppose that the set X is a subset of Y . Find
(a) X ∪ Y
(b) X ∩ Y
(c) X \ Y
2.4 Find the power set of the following sets:
(a) A = {∅, {∅}}.
(b) B = {x, y, z}.
2.5 Determine if the following are elements of the set A × B × C where A =
{0, 1, 2}, B = {1, 2}, and C = {0, 1}.
(a) {∅}.
(b) {1}.
(c) (0, 2, 1).
(d) (2, 2).
(e) {(1, 1, 1)}
(f) (1, 0, 1).
2.6 Given |A| = m, |B| = n and |C| = k determine the lower and upper
bounds for |(A × B) \ C|.
2.7 List all elements of the set {x ∈ R : 2x2 − x − 7 = 0}.
2.8 Is the statement true or false? Explain.
(a) ∃m ∈ (0, 1] ∀x ∈ (0, 1] (m ≤ x).
(b) ∀x ∈ (0, 1] ∃y ∈ (0, 1] (x < y).
(c) ∀x ∈ (0, 1] ∃y ∈ (0, 1] (y < x).
(d) ∃M ∈ (0, 1] ∀x ∈ (0, 1] (x ≤ M ).
2.9 Is the statement true or false? Explain.
(a) ∀x ∈ Z ∀y ∈ Z ∃z ∈ Z (x < y → x < z < y)
(b) ∀x ∈ Q ∀y ∈ Q ∃z ∈ Q (x < y → x < z < y)
2.10 Given intervals of real numbers A = [1, 10], B = (10, 20], C = (5, 20).
Then A \ C can be expressed as {x ∈ R : 1 ≤ x ≤ 5} or [1, 5]. Write the
following sets in similar forms, if possible.
(a) A \ B.
(b) B \ C.
(c) A ∩ B.
(d) C \ B.
2. SET THEORY
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Medium
2.11
2.12
2.13
2.14
2.15
2.16
2.17
2.18
2.19
2.20
2.21
2.22
2.23
Prove that the empty set, ∅, is a subset of every set.
Let A = {a ∈ R : ∀x ∈ R (x2 + 6x ≥ a)}. Describe A.
Let C = {c ∈ R : ∀x ∈ R (x2 + 6x ≤ c)}. Describe C.
Let B = {b ∈ R : ∃x ∈ R (2x2 − bx + 3 = 0)}. Describe B.
Determine which of the following statements are true in the case of three
arbitrary sets P , Q, and R.
(a) If P is an element of Q and if Q is a subset of R, then P is an element
of R.
(b) If P is an element of Q and if Q is a subset of R, then P is also a
subset of R.
(c) If P is a subset of Q and Q is an element of R, then P is an element
of R.
(d) If P is a subset of Q and Q is an element of R, then P is a subset of
R.
Does A \ B = C imply A = B ∪ C? Prove your answer.
Does A = B ∪ C imply A \ B = C? Prove your answer.
Let A 6= ∅. Does A × B = A × C imply B = C? Prove your answer. What
if A = ∅?
Prove or disprove the following assertions involving three arbitrary sets
A, B, and C.
(a) (A \ B) \ C = A \ (B ∪ C).
(b) (A \ B) \ C = (A \ B) \ (B \ C).
Prove (A \ B) ∪ (B \ A) = (A ∪ B) \ (A ∩ B) for every two sets A and B.
Prove that P (A) ∩ P (B) is equal to P (A ∩ B) for every two two sets A
and B. What can we say if the intersection operation, ∩, is replaced by
the union operation, ∪, in the above?
If the symmetric difference of sets A and B is equal to the symmetric
difference of the sets A and C, is it necessary that B = C? Explain your
answer.
Let Ai = {1, 2, 3, . . . , i} for i = 1, 2, 3, . . .. Find
n
n
\
[
Ai and
Ai .
i=1
i=1
2.24 Let O be a point of a plane α. For each positive real number r, let
C(O, r) = {P ∈ α : OP = r}.
What geometrical figure C(O, r) is? Describe
[
C(O, r).
r∈R,r>0
2.25 Let A = {1, 2, 3, . . . , 30}. For each integer i ≥ 2, define
Xi = {ik : k ∈ N and k ≥ 2}.
Describe the set
A\
30
[
i=2
Xi .
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2. SET THEORY
2.26 Prove the following assertions involving three arbitrary sets A, B, and C.
(a) A × (B ∩ C) = (A × B) ∩ (A × C).
(b) (A ∪ B) × C = (A × C) ∪ (B × C).
2.27 A barber in an army was given an order to shave those and only those
men who do not shave themselves. Should he shave himself?
(This famous paradox shows that not every property can be used to
define a set.)
2.28 (a) What the union of the sets of points all squares (with their interiors)
inscribed in a given circle?
(b) What the intersection of the sets of points of all squares (with their
interiors) inscribed in a given circle?
(c) What the union of the sets points of all squares (with their interiors)
circumscribed around a given circle?
(d) What the intersection of the sets points of all squares (with their
interiors) circumscribed around a given circle?
2.29 Solve previous problem where all squares with interiors are replaced by
squares without their interiors.
2. SET THEORY
11
Some Answers and Hints to Set Theory Section
Note that what is below are just answers and hints. More details are needed (at
least to some of them) when you present solutions of the problems.
2.1. (a) T ; F ; T; T. (b) T; F; F; T. (c) F; T. (d) T; T. (e) F; T. (f) T
2.2. E.g., X3 ∩ X5 = ∅, X2 ∩ X4 = ∅, X2 ∩ X2 = X2 . There are 15 cases to consider.
2.3. (a) Y ; (b) X; (c) ∅.
2.4. (a) Hint: it has 4 elements; (b) Hint: it has 8 elements.
2.5. (a), (b), (d), (e), (f) – No. (c) – Yes.
2.6. The lower bound is 0, the upper is mn. This means that we will always have:
0 ≤ |(A × B) − C| ≤ mn.
Hint: Think what C can be. Give examples when the number of elements is 0, and
when it is
√ mn. √
2.7. { 1−4 57 }, 1+4 57 }.
2.8. (a) F; (b) F; (c) T; (d) T.
2.9. (a) F; (b) T.
2.10. (a) {x ∈ R : 1 ≤ x ≤ 10}, or [1, 10]. Both are equal to A. (b) {20} (or [20, 20],
but this notation is rarely used. (c) ∅. (d) {x ∈ R : 5 < x ≤ 10}, or (5, 10].
2.12. a ≤ −9, or, equivalently, (−∞, −9]. Hint: Complete the square in x2 + 6x,
or find the y-coordinate of the vertex of the parabola y = x2 + 6x.
2.13. C = ∅.
√
√
2.14. B = (−∞, −2 6] ∪ [2 6, ∞).
2.15. (a) T; (b), (c), (d) – F. Give a counterexample to each false statement.
2.16. No. Give a counterexample.
2.17. No. Give a counterexample.
2.18. Yes. If A = ∅, the answer is No.
2.19. Can use either the method presented in the text, or the one presented in
class.
2.21. Not true for the union. Give a counterexample.
2.22. No. Find a counterexample.
2.23. An = {1, 2, . . . , n}, and {1}.
2.24. The whole plane with point O removed.
2.25 Hint: there are 11 numbers in the set.
2.27 The decision cannot be made.
2.28 Answers: (a) the disc bounded by the circle. (b) a disc. Find its radius. (c) a
disc. Find its radius. (d) the disc bounded by the circle.
2.29 Answers: (a) concentric annulus. Find its radii. (b) empty set (c) concentric
annulus. Find its radii. (d) empty set.