Math 301 - Fundamentals of Mathematics - Journal 1 Problem List
(A) Let P and Q be statements for which P ⇒ Q is true and ¬Q is true. Prove (or disprove and salvage)
the following statements:
(a) P is true.
(b) P ∧ Q is true.
(c) P ∨ Q is true.
(B) Let P and Q be statements for which P ⇒ Q is false. Prove (or disprove and salvage) the following
statements:
(a) (¬P ) ⇒ (¬Q) is true.
(b) Q ⇒ P is true.
(c) P ∧ Q is true.
(C) Let P , Q, and R be statements for which Q is false and (¬P ) ⇒ Q is true. Prove (or disprove and
salvage) the following statements:
(a) (¬Q) ⇒ P is true.
(b) P is true.
(c) P ∧ R is true.
(d) R ⇒ (¬P ) is true.
(D) Suppose each of the following statements is true:
• Laura is in the seventh grade.
• Laura got an A on the mathematics test or Sarah got an A on the mathematics test.
• If Sarah got an A on the mathematics test, then Laura is not in the seventh grade.
Prove (or disprove and salvage) the following statements:
(a) Laura got an A on the mathematics test.
(b) Sarah got an A on the mathematics test.
(c) Either Laura or Sarah did not get an A on the mathematics test.
(E) A tautology is a compound statement that is true for all possible truth values of the components
of the statement. For example, P ∨ (¬P ) is a tautology. A contradiction is a compound statement
that is false for all possible truth values of the components of the statement. For example, P ∧(¬P ) is
a contradiction. For each of the following statements, determine if it is a tautology, a contradiction,
or neither and prove your answers.
(a) (¬Q) ∧ (P ⇒ Q).
(b) Q ∧ [P ∧ (¬Q)].
(c) (Q ∧ P ) ∧ [P ⇒ (¬Q)].
(d) (¬Q) ⇒ (P ∧ ¬P ).
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(F) A tautology is a compound statement that is true for all possible truth values of the components
of the statement. For example, P ∨ (¬P ) is a tautology. A contradiction is a compound statement
that is false for all possible truth values of the components of the statement. For example, P ∧ (¬P )
is a contradiction. Prove (or disprove and salvage) the following statements:
(a) [(P ⇒ Q) ∧ P ] ⇒ Q is a tautology.
(b) [(P ⇒ Q) ∧ (Q → R)] ⇒ (P → R) is a tautology.
(G) Suppose that P and Q are true statements, U and V are false statements and the truth value of the
statement W is unknown. For each of the following statements, determine which are true, which are
false, and which it is impossible to determine their truth. Make sure to prove your answers.
(a) (P ∨ Q) ∨ (U ∧ W )
(f) [(¬P ) ∨ (¬U )] ∧ [Q ∨ (¬V )]
(b) P ∧ (Q ⇒ W )
(g) [P ∧ (¬V )] ∧ (U ∨ W )
(c) P ∧ (W ⇒ Q)
(h) [P ∨ (¬Q)] ⇒ (U ∧ W )
(d) W ⇒ (P ∧ U )
(i) (P ∨ W ) ⇒ (U ∧ W )
(e) W ⇒ [P ∧ (¬U )]
(j) [U ∧ (¬V )] ⇒ (P ∧ W )]
(H) Prove (or disprove and salvage) the following statements:
(a) The two statements below are logically equivalent:
• [(P ∨ Q) ⇒ R]
• (P ⇒ R) ∧ (Q ⇒ R)
(b) The two statements below are logically equivalent:
• (P ∨ Q) ∧ ¬(P ∧ Q)
• [P ∧ (¬Q)] ∨ [Q ∧ (¬P )]
(I) Consider the statement “If xy is even, then x is even or y is even.” We can write this in symbolic
form as
P ⇒ (Q ∨ R)
where P is the statement “xy is even”, Q is the statement “x is even”, and R is the statement “y is
even”.
(a) Write the symbolic form of the contrapositive of P ⇒ (Q ∨ R).
(b) Use DeMorgan’s Law’s to rewrite the hypothesis of the conditional statement obtained in part
(a).
(c) Rewrite your statement from part (b) in English with as few symbols as possible.
(d) Prove your statement from part (c).
(e) What does this tell you about the truth of the original statement for this problem?
(J) Prove (or disprove and salvage) the following statements:
(a) If m is an even integer, then m + 1 is an odd integer.
(b) If m is an odd integer, then m + 1 is an even integer.
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(K) Prove (or disprove and salvage) the following statements:
(a) If x is an even integer and y is an even integer, then x + y is an even integer.
(b) If x is an even integer and y is an odd integer, then x + y is an odd integer.
(c) If x is an odd integer and y is an odd integer, then x + y is an even integer.
(L) Prove (or disprove and salvage) the following statements:
(a) If m is an even integer and n is an integer, then mn is an even integer.
(b) If n is an even integer, then n2 is an even integer.
(c) If n is an odd integer, then n2 is an odd integer.
(M) Prove (or disprove and salvage) the following statements:
(a) If m is an even integer, then 5m + 7 is an odd integer.
(b) If m is an odd integer, then 5m + 7 is an even integer.
(c) If m and n are odd integers, then mn + 7 is an even integer.
(N) Prove (or disprove and salvage) the following statements:
(a) If m is an even integer, then 3m2 + 2m + 3 is an odd integer.
(b) If m is an odd integer, then 3m2 + 7m + 12 is an even integer.
(O) Prove (or disprove and salvage) the following statements:
(a) If a, b, and c are integers, then ab + ac is an even integer.
(b) If b and c are odd integers, and a is an integer, then ab + ac is an even integer.
(P) An integer a is said to be a type 0 integer if there exists an integer n such that a = 3n. An integer
a is said to be a type 1 integer if there exists an integer n such that a = 3n + 1. An integer a is
said to be a type 2 integer if there exists an integer n such that a = 3n + 2.
(a) Give examples of at least four different integers of each type.
(b) By doing examples, does it appear that the following statement is true or false?
If a and b are both type 1 integers, then ab is a type 1 integer.
(c) Prove (or disprove an salvage) the statement in part (b).
(Q) An integer a is said to be a type 0 integer if there exists an integer n such that a = 3n. An integer
a is said to be a type 1 integer if there exists an integer n such that a = 3n + 1. An integer a is
said to be a type 2 integer if there exists an integer n such that a = 3n + 2. Prove (or disprove
and salvage) the following statements:
(a) If a and b are type 1 integers, then a + b is a type 2 integer.
(b) If a and b are type 2 integers, then a + b is a type 1 integer.
(c) If a is a type 1 integer and b is a type 2 integer, then ab is a type 2 integer.
(d) If a and b are type 2 integers, then ab is a type 1 integer.
(R) Prove (or disprove and salvage) the following statements:
(a) For all integers a, b, and c with a 6= 0, if a | b and a | c, then a | (b − c).
(b) For each n ∈ Z, if n is an odd integer, then n3 is an odd integer.
(c) For each integer a, if 4 divides (a − 1), then 4 divides (a2 − 1).
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(S) For each of the following statements, use a counterexample to prove it is false. In each case, the
quantifier used is the universal quantifier. Would changing the quantifier to an existential quantifier
change the truth of the statement?
(a) For each odd natural number n, if n > 3, then 3 divides (n2 − 1).
(b) For each natural number n, the number 3 · 2n + 2 · 3n + 1 is prime.
p
(c) For all real numbers x and y, x2 + y 2 > 2xy.
(d) For each integer a, if 4 divides (a2 − 1), then 4 divides (a − 1).
(T) Prove (or disprove and salvage) the following statements:
(a) For all integers a, b, and c, with a 6= 0, if a | b, then a | (bc).
(b) For all integers a and b with a 6= 0, if 6 | (ab), then 6 | a or 6 | b.
(c) For all integers a, b, and c, with a 6= 0, if a divides (b − 1) and a divides (c − 1), then a divides
(bc − 1).
(d) For each integer n, if 7 divides n2 − 4, then 7 divides n − 2.
(U) Prove (or disprove and salvage) the following statements:
(a) For every integer n, the integer 4n2 + 7n + 6 is odd.
(b) For every odd integer n, the integer 4n2 + 7n + 6 is odd.
(c) For all integers a, b, and d, with d 6= 0, if d divides both (a − b) and (a + b), then d divides a.
(d) For all integers a, b, and c, with a 6= 0, if a | (bc), then a | b or a | c.
(V) Prove (or disprove and salvage) the following statements:
(a) If x and y are integers and xy = 1, then x = 1 or x = −1.
(b) For all nonzero integers a and b, if a | b and b | a, the a = ±b.
(W) Prove (or disprove and salvage) the following statements:
(a) Let a be an integer. If there exists an integer n such that a | (4n + 3) and a | (2n + 1), then
a = 1 or a = −1.
(b) For each integer a, if there exists an integer n such that a divides (8n + 7) and a divides (4n + 1),
then a divides 5.
(c) For each integer a, if there exists an integer n such that a divides (9n + 5) and a divides (6n + 1),
then a divides 7.
(X) Prove (or disprove and salvage) the following statements:
(a) For each integer n, if n is odd, then 8 divides (n4 + 4n2 + 11).
(b) For each integer n, if n is odd, then 8 divides (n4 + n2 + 2n).
(Y) Let n be an integer. Prove (or disprove and salvage) the following statements:
(a) If n is even, then n3 is even.
(b) If n3 is even, then n is even.
(c) The integer n is even if and only if n3 is an even integer.
(d) The integer n is odd if and only if n3 is an odd integer.
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(Z) Prove (or disprove and salvage) the following statements:
(a) For all integers a and b, if ab is even, then a is even or b is even.
(b) For each integer n, n is even if and only if 4 divides n2 .
(c) For each integer a, if a2 − 1 is even, then 4 divides a2 − 1.
(AA) Let a and b be natural numbers such that a2 = b3 . Prove (or disprove and salvage) the following
statements:
(a) If a is even, then 4 divides a.
(b) If 4 divides a, then 4 divides b.
(c) If 4 divides b, then 8 divides a.
(d) If a is even, then 8 divides a.
(e) There are no examples of natural numbers a and b such that a is even and a2 = b3 , but b is not
divisible by 8.
(AB) Prove (or disprove and salvage) the following statements:
(a) For all integers a and b, if a is even and b is odd, then 4 does not divide (a2 + b2 ).
(b) For all integers a and b, if a is even and b is odd, then 6 does not divide (a2 + b2 ).
(c) For all integers a and b, if a is even and b is odd, then 4 does not divide (a2 + 2b2 ).
(d) For all integers a and b, if a is odd and b is odd, then 4 divides (a2 + 3b2 ).
(AC) Consider the following statement: “There are no integers a and b such that b2 = 4a + 2.
(a) Rewrite the statement in an equivalent form using a universal quantifier.
(b) Prove your statement from part (a).
(AD) For each of the following, find a counterexample to show the statement is false. Then write the
negation of the statement in English, without using symbols for quantifiers.
(a)
(b)
(c)
(d)
(e)
∀m ∈ Z, m2 is even.
∀x ∈ R, x2 > 0.
∃x ∈ Q, x2 − 3x − 7 = 0.
√
∀x ∈ R, x ∈ R.
∀m ∈ Z, m
3 ∈ Z.
(f) ∃x ∈ R, x2 + 1 = 0.
√
(g) ∀a ∈ Z, a2 = a.
(h) ∀x ∈ R, tan2 x + 1 = sec2 x.
(i) ∃m ∈ N, m2 < 1.
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(AE) These problems contain definitions or results from more advanced classes. Even though you will
not understand all of the terms involve, it is still possible to recognize the structure of the given
statements and write meaningful negations.
(a) In abstract algebra, an operation ∗ on a set A is called a commutative operation provided
that for all a, b ∈ A, a ∗ b = b ∗ a. Carefully explain what it means to say that an operation ∗
on a set A is not a commutative operation.
(b) In abstract algebra, a ring consists of a nonempty set R and two operations called addition and
multiplication. A nonzero element r in a ring R is called a zerodivisor if there exists a nonzero
element t in R such that rt = 0. Carefully explain what it means to say that a nonzero element
r in a ring R is not a zerodivisor.
(c) A set M of the real numbers is called a neighborhood of a real number r if there exists a
positive real number ε such that the open interval (r − ε, a + ε) is contained in M . Carefully
explain what it means to say that a set M is not a neighborhood of a real number a.
(d) In analysis, a sequence of real numbers, {x1 , x2 , x3 , . . . , xk , . . .} is called a Cauchy sequence
provided that for each positive real number ε, there exists a natural number N such that for
all m, n ∈ N, if m > N and n > N , then |xm − xn | < ε. Carefully explain what it means to say
that a sequence or real numbers {x1 , x2 , x3 , . . . , xk , . . .} is not a Cauchy sequence.
(AF) Consider the statement: “For all integers a and b, if 3 does not divide a and 3 does not divide b, then
3 does not divide the product ab.”
(a) Notice that the hypothesis of this statement is stated as a conjunction of two negatives. Also, the
conclusion is stated as the negation of a statement. This often indicates that we should consider
using a proof by the contrapositive. Using the symbolic form of our statement, [(¬P ) ∧ (¬Q)] ⇒
(¬R), what are the statements P , Q, and R?
(b) Write a symbolic form for the contrapositive of [(¬P ) ∧ (¬Q)] ⇒ (¬R).
(c) Write the contrapositive of the statement as a conditional statement in English.
(AG) This journal problem is intended to provide another rationale as to why a proof by contradiction
works. Suppose we are trying to prove that a statement P is true. Instead of proving this statement,
suppose we prove the conditional statement (¬P ) ⇒ C where C is some contradiction. Recall that
a contradiction is a statement that is always false.
(a) In symbols, write a statement that is a disjunction and is logically equivalent to (¬P ) ⇒ C.
(b) Since we have proven (¬P ) ⇒ C is true, then the disjunction from part (a) must also be true.
Use this to explain why the statement P must be true.
(c) Now, explain why P must be true if we prove that the negation of P implies a contradiction.
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(AH) One possible definition of a prime number is that a natural number p is a prime number if it is
greater than 1 and the only natural number factors of p are 1 and p. A natural number other than 1
that is not a prime number is called a composite number. The natural number 1 is neither prime
nor composite. Using this definition, we see that 2, 3, 5, and 7 are prime numbers and 4, 10, and 60
are composite numbers since 4 = 2 · 2, 10 = 2 · 5, and 60 = 5 · 12. Use the given definitions of prime
and composite numbers to do the following exercises.
(a) Give examples of four natural numbers other than 2, 3, 5, and 7 which are prime numbers.
(b) Explain why a natural number p that is greater than 1 is a prime number given that: “For all
d ∈ N, if d is a factor of p, then d = 1 or d = p.”
(c) Give examples of four natural numbers other than 4, 10, and 60 that are composite numbers
and explain why they are composite numbers.
(d) Write a useful description of what it means to say that a natural number is a composite number
(other than saying that it is not prime).
(AI) Three natural numbers a, b, and c with a < b < c are said to form a Pythagorean triple if
a2 + b2 = c2 . For example, 3, 4, and 5 form a Pythagorean triple since 32 + 42 = 52 . The study
of Pythagorean triples began with the development of the Pythagorean theorem for right triangles,
which states that if a and b are the lengths of the legs of a right triangle and c is the hypotenuse,
then a2 + b2 = c2 . For example, if the lengths of the legs of a right triangle
√ are 4 and 7 units, then
c2 = 42√+ 72 = 16 + 49 = 65, and the length of
the
hypotenuse
must
be
65 units. Notice that 4,
√
7, and 65 are not a Pythagorean triple since 65 is not a natural number. Prove (or disprove and
salvage) the following statements:
(a) The following are Pythagorean triples:
• 3, 4, and 5
• 6, 8, and 10
• 8, 15, and 17
• 10, 24, and 26
• 12, 35, and 37
• 14, 48, and 50
(b) There exists a Pythagorean triple of the form m, m + 7, m + 8. Assuming this statement is true,
find all such Pythagorean triples. If none exists, prove that instead.
(c) There exists a Pythagorean triple of the form m, m + 11, m + 12. Assuming this statement is
true, find all such Pythagorean triples. If none exists, prove that instead.
(AJ) Three natural numbers a, b, and c with a < b < c are said to form a Pythagorean triple if
a2 + b2 = c2 . For example, 3, 4, and 5 form a Pythagorean triple since 32 + 42 = 52 . The study
of Pythagorean triples began with the development of the Pythagorean theorem for right triangles,
which states that if a and b are the lengths of the legs of a right triangle and c is the hypotenuse,
then a2 + b2 = c2 . For example, if the lengths of the legs of a right triangle
√ are 4 and 7 units, then
c2 =√
42 + 72 = 16 + 49 = 65, and the length √
of the hypotenuse must be 65 units. Notice that 4, 7,
and 65 are not a Pythagorean triple since 65 is not a natural number.
(a) Let the smallest even natural number in a Pythagorean triple be called n and let m be such
that n = 2m. Write formulas for the other two numbers in each Pythagorean triple below in
terms of m. Check to see if your formula works by determining whether or not you obtain a
Pythagorean triple using m = 10, i.e. a Pythagorean triple whose smallest even number is 20.
If it fails, try again.
(b) Prove your conjectured formula from part (a) works.
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(AK) For a right triangle, suppose that the hypotenuse has length c feet and the lengths of the sides are a
feet and b feet.
(a) What is the formula for the area of this right triangle? What is an isosceles triangle?
(b) State the Pythagorean Theorem for right triangles.
(c) Prove that the right triangle described above is an isosceles triangle if and only if the area of
the right triangle is 14 c2 .
(AL) Prove (or disprove and salvage) the following statement: For all integers a and m, if a and m are the
lengths of the sides of a right triangle, and m + 1 is the length of the hypotenuse, then a is an odd
integer.
(AM) Prove (or disprove and salvage) the following statements:
(a) There exist integers x and y such that 4x + 6y = 2.
(b) There exist integers x and y such that 6x + 15y = 2.
(c) There exist integers x and y such that 6x + 15y = 9.
(AN) A magic square is a square array of natural numbers whose rows, columns, and diagonals all sum to
the same number. For example, the following is a 3 × 3 magic square since the sum of the 3 numbers
in each row, each column, and each diagonal are all 15.
8
1
6
3
5
4
4
9
2
Prove that the following 4 × 4 square cannot be completed to form a magic square.
3
6
9
1
4
7
2
5
8
10
Hint: Assign each of the six blank cells in the square a name. One possibility is to use a, b, c, d, e,
and f .
(AO) A magic square is a square array of natural numbers whose rows, columns, and diagonals all sum
to the same number. Is it possible to construct a 3 × 3 magc square using the digits 1 through 9
each only once with the digit 3 in the center? That is, is it possible to complete the following magic
square using the digits 1, 2, 4, 5, 6, 7, 8, 9 once each? Either construct one, or prove that no such
magic square is possible.
3
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(AP) If we calculate successive powers of 2, i.e. compute 21 , 22 , 23 , 24 , 25 , etc., and examine the units
digits of these numbers, we could make the following conjectures (among others):
• If n is a natural number, then the units digit of 2n must be 2, 4, 6, or 8.
• The units digits of the successive powers of 2 repeat according to the pattern “2, 4, 6, 8.”
(a) Formulate a conjecture about the units digits of successive powers of 4, i.e. 41 , 42 , 43 , 44 , 45 ,
etc.
(b) Is it possible to formulate a conjecture about the units digit of numbers of the form 7n −2n where
n is a natural number. If so, formulate a conjecture in the form of a conditional statement.
(c) Let f (x) = e2x . Determine the first eight derivatives of this function. What do you observe?
Formulate a conjecture that appears to be true, and write it in the form of a conditional
statement.
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