- Information
- AI Chat
Book notes
Mathematics (MATH 271)
University of Calgary
Preview text
Chapter 1 SPEAKING
MATHEMATICALLY
Universal Statement: A certain property is true for all elements in a set. Ex. All positive numbers are greater than zero.
Conditional Statement: If one thing is true, then some other thing also has to be true. Ex. If 378 is divisible by 18, then 378 is divisible by 6.
Existential Statement: There is at least one thing for which the property is true. Ex. There is a prime number that is even.
Universal Conditional Statements: A statement that is both universal AND conditional which can be rewritten in purely universal or purely conditional format. Ex. For all animals a, if A is a dog, then A is a mammal. Implicit in universal and explicit in conditional: If A is a dog, then A is a mammal. If an animal is a dog, then the animal is a mammal. Implicit in conditional and explicit in universal: For all dogs A, A is a mammal. All dogs are mammal.
Universal Existential Statement: A statement that is universal because its first part says that a certain property is true for all objects of a given type, and it is existential because its second part asserts the existence of something. Ex. Every real number has an additive inverse. Universal: “has an additive inverse” applies to each and every real number out there. Existential: “has an additive inverse” asserts the existence of “an additive inverse”. All real numbers has additive inverses. For all real numbers r, there is an additive inverse for r. For all real numbers r, there is a real number s such that s is an additive inverse for r.
Existential Universal Statement: A statement that is existential because its first part asserts that a certain object exists and is universal because its second part says that the object satisfies a certain property for all things of a certain kind.
Ex. There is a positive integer that is less than or equal to every positive integer. Existential: “There is a positive integer” asserts the existence of a positive integer. Universal: “Every positive integer”. Some positive integer is less than or equal to every positive integer. There is a positive integer m that is less than or equal to every positive integer. There is a positive integer m such that every positive integer i s greater than or equal to m. There is a positive integer with the property that for all positive integers n, m ≤n.
If S is a set, the notationx∈Smeans that x is an element of S.
The notation x∉S means that x is not an element of S.
Symbol Set R Set of all real numbers. Z Set of all integers. Q Set of all rational numbers, or quotients of integers.
Let S denote a set and let P(x) be a property that elements of S may or may not satisfy. We may define a new set to be the set of all elements x in S such that P(x) is true. We denote this set as follows: {x ∈ S | P(x)}
If A and B are sets, then A is called a subset of B, written A ⊆ B, if, and only if, every element of A is also an element of B. A ⊆ B means that: For all elements x, if x ∈ A then x ∈ B. The phrases A is contained in B and B contains A are alternative ways of saying that A is a subset of B. A∉B means that there is at least one element x such that x∈A∧x∉B.
A is a proper set of B if, and only if, every element of A is in B but there is at least one element of B that is not in A.
Given elements a and b, the symbol (a, b) denotes the ordered pair consisting of a and b together with the specification that a is the first element of the pair and b is the
Exercise Set 1.
Is there a real number whose square is −1? a) Is there a real number x such that x 2 is -1? b) Does there exist a real number x such that x 2 = −1?
Is there an integer that has a remainder of 2 when it is divided by 5 and a remainder of 3 when it is divided by 6? a) Is there an integer n such that n has a remainder of when n is divided by 5 and a remainder of 3 when n is divided by 6? b) Does there exist an integer n such that if n is divided by 5 the remainder is 2 and if divided by 6?
Given any two real numbers, there is a real number in between. a) Given any two real numbers a and b, there is a real number c such that c is between the real numbers a & b. b) For any two real number a & b, there exists a real number c, such that a < c < b.
Given any real number, there is a real number that is greater. a) Given any real number r, there is a real number s such that s is greater than r. b) For any real number s, there exists a real number r, such that s > r.
The reciprocal of any positive real number is positive. a) Given any positive real number r, the reciprocal of r is also positive. b) For any real number r, if r is positive, then the reciprocal of r is also positive. c) If a real number r is positive, then 1/r is also positive.
The cube root of any negative real number is negative. a) Given any negative real number s, the cube root of s is also negative.
b) For any real number s, if s is negative, then the cube root of s is also negative. c) If a real number s is negative, then the cube root of s is also negative.
- Rewrite the following statements less formally, without using variables. Determine, as best as you can, whether the statements are true or false. a) There are real numbers u and v with the property that u +v < u − v. The difference of any two real numbers is greater than the sum of that two numbers. b) There is a real number x such that x 2 < x. For any real numbers, the real number itself is greater than the real number squared. c) For all positive integers n, n 2 ≥ n. The square of any positive integer is greater than or equal to the integer itself. d) For all real numbers a and b, |a + b| ≤ |a| + |b|. The sum of the absolute value of a real number with another real number is greater than or equal to the absolute value of the sum of the two real numbers.
In each of 8–13, fill in the blanks to rewrite the given statement. 8. For all objects J, if J is a square then J has four sides. a) All squares have 4 sides. b) Every square has 4 sides. c) If an object is a square, then it has 4 sides. d) If J is a square, then J has 4 sides. e) For all squares J, J have 4 sides.
- For all equations E, if E is quadratic then E has at most two real solutions. a) All quadratic equations have two real solutions at most. b) Every quadratic equation has two real solutions at most. c) If an equation is quadratic, then it has two real solutions at most. d) If E is quadratic, then E has two real solutions at most. e) For all quadratic equations E, E have two real solutions at most.
10 nonzero real number has a reciprocal. a) All nonzero real numbers have a reciprocal. b) For all nonzero real numbers r, there is a reciprocal for r. c) For all nonzero real numbers r, there is a real number s such that s is the reciprocal of r.
11 positive number has a positive square root. a) All positive numbers have positive square roots. b) For any positive number e, there is a positive square root for e.
b) 1 c) 2 d) Yes e) No
Set A = Set D Set B = Set C
T 2 = {2, 4} T-3 = {-3, 9} T 1 = {1, 1} T 0 = {0, 0}
a) S = {-1, 1} b) T = {0, 2} c) U = {D.N} d) V = {D.N} e) W = {D.N} f) X = {.., 2, 3, 4...}
a) No b) Yes c) Yes d) Yes
a) Yes b) No c) No d) Yes e) Yes f) Yes g) Yes h) No i) Yes j) Yes
No b) No c) Yes d) Yes
A×B={(w,a),(w,b),(x,a),(x ,b),(y ,a),(y ,b),(z ,a),(z,b)} b) B× A={(a,w),(a, x),(a, y),(a,z),(b,w),(b, x),(b , y),(b, z)} c) A× A={(w ,w),(w ,x),(w , y),(w , z),(x ,w),(x , x),(x , y),(x, z),(y ,w),(y , x),(y , y),(y ,z),(z,w),(z , x),(
d) B×B={(a,a),(a,b),(b,a),(b,b)}
- S×T={(2,1),(2,3),(2,5),(4,1),(4,3),(4,5),(6,1),(6,3),(6,5)} b) T ×S={(1,2),(1,4),(1,6),(3,2),(3,4),(3,6),(5,2),(5,4),(5,6)}
Exercise Set 1.
- Let A = {2, 3 , 4} and B = {6, 8 , 10} and define a relation R from A to B as follows:
For all (x, y) ∈ A × B, (x, y) ∈ R means that y x is an integer. a) Is 4 R 6? Is 4 R 8? Is ( 3 , 8 ) ∈ R? Is ( 2 , 10 ) ∈ R? No, Yes, No, Yes b) Write R as a set of ordered pairs. R = {(2, 6), (2, 8), (2, 10), (3, 6), (4, 8)} c) Write the domain and co-domain of R. Domain: A = {2, 3, 4} Co-domain: B = {6, 8, 10}
- Let C = D = {−3,−2,−1, 1 , 2 , 3} and define a relation S from C to D as follows: For
all (x, y) ∈ C × D, (x, y) ∈ S means that 1 x
− 1
y is an integer. a) Is 2 S 2? Is −1S − 1? Is ( 3 , 3 ) ∈ S? Is ( 3 ,−3) ∈ S? Yes, Yes, Yes, No b) Write S as a set of ordered pairs. S = {(-3, -3), (-2, -2), (-1, -1), (1, 1), (2, 2), (3, 3)} c) Write the domain and co-domain of S. Domain: C = {− 3 , − 2 , − 1 , 1 , 2 , 3 } Co-domain: D = {− 3 , − 2 , − 1 , 1 , 2 , 3 }
∧ q. It is true when, and only when, both p and q are true. If either p or q is false, or if both are false, p ∧ q is false. P Q P∧Q True True True False True False True False False False False False
The “or” in math is INCLUSIVE.
If p and q are statement variables, the disjunction of p and q is “p or q,” denoted p ∨ q. It is true when either p is true, or q is true, or both p and q are true; it is false only when both p and q are false. P Q P∨Q True True True False True True True False True False False False
Two statement forms are called logically equivalent if, and only if, they have identical truth values for each possible substitution of statements for their statement variables. De Morgan’s Laws The negation of an and statement is logically equivalent to the or statement in which each component is negated. The negation of an or statement is logically equivalent to the and statement in which each component is negated. Not (P and Q) = not P or not Q Not (P or Q) = not P and not Q
- Commutative laws: p ∧ q ≡ q ∧p p∨ q ≡ q ∨ p
- Associative laws: (p ∧ q) ∧ r ≡ p ∧ (q ∧ r) (p ∨ q) ∨ r ≡ p ∨ (q ∨ r )
- Distributive laws: p ∧ (q ∨ r ) ≡ (p ∧ q) ∨ (p ∧ r ) p ∨ (q ∧ r ) ≡ (p ∨ q) ∧ (p ∨ r )
- Identity laws: p ∧ t ≡p p∨ c ≡ p
- Negation laws: p ∨ ∼p ≡ t p ∧ ∼p ≡ c
- Double negative law: ∼(∼p) ≡ p
- Idempotent laws: p ∧ p ≡p p∨ p ≡ p
- Universal bound laws: p ∨ t ≡ t p ∧ c ≡ c
- De Morgan’s laws: ∼(p ∧ q) ≡ ∼p ∨ ∼q ∼(p ∨ q) ≡ ∼p ∧ ∼q
- Absorption laws: p ∨ (p ∧ q) ≡p p∧ (p ∨ q) ≡ p
- Negations of t and c: ∼t ≡ c ∼c ≡ t
- A tautology is a statement form that is always true regardless of the truth values of
the individual statements substituted for its statement variables.
A contradiction is a statement form that is always false regardless of the truth values of the individual statements substituted for its statement variables.
If p and q are statement variables, the conditional of q by p is “If p then q” or “p implies q” and is denoted p →q. It is false when p is true and q is false; otherwise it is true. We call p the hypothesis (or antecedent) of the conditional and q the conclusion (or consequent). A conditional statement that is true by virtue of the fact that its hypothesis is false is often called vacuously true or true by default. Thus the statement “If you show up for work Monday morning, then you will get the job” is vacuously true if you do not show up for work Monday morning. In general, when the “if” part of an if-then statement is false, the statement as a whole is said to be true, regardless of whether the conclusion is true or false. Consider the statement: If 0 = 1 then 1 = 2. Since the hypothesis of this statement is false, the statement as a whole is true.
∼(p →q) ≡ p ∧ ∼q
The negation of “if p then q” is logically equivalent to “p and not q.”
The contrapositive of a conditional statement of the form “If p then q” is If ∼q then ∼p. Symbolically, The contrapositive of p →q is ∼q →∼p. A conditional statement is logically equivalent to its contrapositive. Ex. If P then Q: If one becomes the president, then he must be 50 years old or older. Contrapositive: If ~Q then ~P: If one is not 50 years old or older, then he cannot become the president. P only if Q：One becomes the president only if he is 50 years old or older.
Suppose a conditional statement of the form “If p then q” is given.
- The converse is “If q then p.” If one is 50 years old or older, then he becomes the president.
- The inverse is “If ∼p then ∼q.”
- If one’s not the president, then he must not be 50 years old or older.
Symbolically, the converse of p →q is q → p, and the inverse of p →q is ∼p →∼q. A conditional statement and its converse are not logically equivalent. A conditional statement and its inverse are not logically equivalent. The converse and the inverse of a conditional statement are logically equivalent to each other.
Exercise Set 2.
a. If all integers are rational, then the number 1 is rational. All integers are rational. Therefore, the number 1 is rational. b. If all algebraic expressions can be written in prefix notation, then. . Therefore, (a + 2b)(a2 − b) can be written in prefix notation.
a. If all computer programs contain errors, then this program contains an error. This program does not contain an error. Therefore, it is not the case that all computer programs contain errors. b. If , then. 2 is not odd. Therefore, it is not the case that all prime numbers are odd.
a. This number is even or this number is odd. This number is not even. Therefore, this number is odd. b. or logic is confusing. My mind is not shot. Therefore,.
a. If n is divisible by 6, then n is divisible by 3. If n is divisible by 3, then the sum of the digits of n is divisible by 3. Therefore, if n is divisible by 6, then the sum of the digits of n is divisible by 3. (Assume that n is a particular, fixed integer.) b. If this function is then this function is differentiable. If this function is then this function is continuous. Therefore, if this function is a polynomial, then this function.
Indicate which of the following sentences are statements. a. 1,024 is the smallest four-digit number that is a perfect square. b. She is a mathematics major. c. 128 = 26 d. x = 26 Write the statements in 6–9 in symbolic form using the symbols ∼,∨, and ∧ and the indicated letters to represent component statements.
Let Q(x) be a predicate and D the domain of x. An existential statement is a statement of the form “∃x ∈ D such that Q(x).” It is defined to be true if, and only if, Q(x) is true for at least one x in D. It is false if, and only if, Q(x) is false for all x in D.
Universal conditional statement: ∀x, if P(x) then Q(x).
∀ x ∈ U , if P ( x) then Q( x ) can always be rewritten in the form ∀ x ∈ D , Q ( x ) by narrowing U to be the domain D consisting of all values of the variable x that make P(x) true. All squares are rectangles. ∀x, if____________, then__________.: For all x, if x is a square, then x is rectangle. ∀_______x, _______.: For all square x, x is rectangles.
“∃x such that p(x) and Q(x)” can be rewritten as “∃xεD such that Q(x),” where D is the set of all x for which P(x) is true. “There is an integer that is both prime and even.”Let Prime (n) be “n is prime” and Even (n) be “n is even.” ∃n such that ∧ .: ∃n such that Prime(n)∧ Even(n). ∃n such that_________. ∃ a prime number n such that Even (n). OR ∃ an even number n such that Prime (n).
Let P(x) and Q(x) be predicates and suppose the common domain of x is D.
The notation P(x) ⇒ Q (x) means that every element in the truth set of P(x) is in the truth set of Q(x), or, equivalently, ∀x, P(x) → Q(x).
The notation P(x) ⇔ Q (x) means that P(x) and Q(x) have identical truth sets, or, equivalently, ∀x, P(x) ↔ Q(x).
The negation of a statement of the form ∀x in D, Q(x) Is logically equivalent to a statement of the form ∃x in D such that ∼Q(x). Symbolically, ∼(∀x ∈ D, Q(x)) ≡ ∃x ∈ D such that ∼Q(x). The negation of a universal statement (“all are”) is logically equivalent to an existential statement (“some are not” or “there is at least one that is not”).
The negation of a statement of the form ∃x in D such that Q(x) Is logically equivalent to a statement of the form ∀x in D,∼Q(x). Symbolically, ∼(∃x ∈ D such that Q(x)) ≡ ∀x ∈ D,∼Q(x).
The negation of an existential statement (“some are”) is logically equivalent to a universal statement (“none are” or “all are not”).
Negation of a Universal Conditional Statement: ∼ (∀x, if P(x) then Q(x)) ≡ ∃x such that P(x) and ∼Q(x).
Universal statements are generalizations of and statements, and existential statements are generalizations of or statements.
A statement of the form ∀x in D, if P(x) then Q(x) is called vacuously true or true by default if, and only if, P(x) is false for every x in D.
There is a bowl sits on a table and next to the bowl is a pile of five blue and five gray balls, any of which may be placed in the bowl. Suppose that no balls at all are placed in the bowl, consider the statement “All the balls in the bowl are blue.” Is this statement true or false? The statement is false if, and only if, its negation is true. And its negation is “There exists a ball in the bowl that is not blue.” But the only way this negation can be true is for there actually to be a non-blue ball in the bowl. And there is not! Hence the negation is false, and so the statement is true “by default.”
- Consider a statement of the form: ∀x ∈ D, if P(x) then Q(x).
- Its contrapositive is the statement: ∀x ∈ D, if ∼Q(x) then ∼P(x).
- Its converse is the statement: ∀x ∈ D, if Q(x) then P(x).
- Its inverse is the statement: ∀x ∈ D, if ∼P(x) then ∼Q(x).
- “∀x, r (x) is a sufficient condition for s(x)” means “∀x, if r (x) then s(x).” “∀x, r (x) is a necessary condition for s(x)” means “∀x, if ∼r (x) then ∼s(x)” or, equivalently, “∀x, if s(x) then r (x).”
- “∀x, r (x) only if s(x)” means “∀x, if ∼s(x) then ∼r (x)” or, equivalently, “∀x, if r (x) then s(x).”
Interpreting Statements with Two Different Quantifiers If you want to establish the truth of a statement of the form ∀x in D, ∃y in E such that P(x, y) Your challenge is to allow someone else to pick whatever element x in D they wish and then you must find an element y in E that “works” for that particular x. If you want to establish the truth of a statement of the form
∀xP(x) → Q(x). Any x that makes P(x) true makes Q(x) true. ∀xQ(x) → R(x). Any x that makes Q(x) true makes R(x) true. ∴ ∀xP(x) → R(x). ∴ Any x that makes P(x) true makes R(x) true.
Chapter 4 ELEMENTARY NUMBER
THEORY AND METHODS OF PROOF
- An integer n is even if, and only of, n equals twice some integer. An integer n is odd if, and only if, n equals twice some integer plus 1. n is even ⇔ ∃an integer k such that n = 2k. n is odd ⇔ ∃an integer k such that n = 2k + 1. Know a particular integer n is evendeduce→ n has the form 2· (some integer).
Know n has the form 2· (some integer)deduce→ n is even.
An integer n is prime if, and only if, n>1 and for all positive integers r and s, if n=rs, then either r or s equals n. An integer n is composite if, and only if, n>1 and n=rs for some integers r and s with 1<r<n and 1<s<n. n is prime ⇔ ∀positive integers r and s, if n = rs then either r = 1 and s = n or r = n and s = 1. n is composite ⇔ ∃positive integers r and s such that n = rs and 1 < r < n and 1 < s < n.
According to the definition given in Section 3, a statement in the form ∃x ∈ D such that Q(x) Is true if, and only if, Q(x) is true for at least one x in D. One way to prove this is to find an x in D that makes Q(x) true. Another way is to give a set of directions for finding such an x. Both of these methods are called constructive proofs of existence.
A nonconstructive proof of existence involves showing either (a) that the existence of a value of x that makes Q(x) true is guaranteed by an axiom or a previously proved theorem or (b) that the assumption that there is no such x leads to a contradiction.
To disprove a statement of the form “∀x ∈ D, if P(x) then Q(x),” find a value of x in D for which the hypothesis P(x) is true and the conclusion Q(x) is false. Such an x is called a counterexample.
To show that every element of a set satisfies a certain property, suppose x is a particular but arbitrarily chosen element of the set, and show that x satisfies the property. Method of Direct Proof
- Express the statement to be proved in the form “∀x ∈ D, if P(x) then Q(x).” (This step is often done mentally.)
- Start the proof by supposing x is a particular but arbitrarily chosen element of D for which the hypothesis P(x) is true. (This step is often abbreviated “Suppose x ∈ D and P(x).”)
- Show that the conclusion Q(x) is true by using definitions, previously established results, and the rules for logical inference.
A real number r is rational if, and only if, it can be expressed as a quotient of two integers with a nonzero denominator. A real number that is not rational is irrational. More formally, if r is a real number, then r is rational ⇔ ∃integers a and b such that r=a b ∧b≠ 0.
- The sum, product, and difference of any two even integers are even.
- The sum and difference of any two odd integers are even.
- The product of any two odd integers is odd.
- The product of any even integer and any odd integer is even.
- The sum of any odd integer and any even integer is odd.
- The difference of any odd integer minus any even integer is odd.
- The difference of any even integer minus any odd integer is odd.
If n and d are integers and d _= 0 then n is divisible by d if, and only if, n equals d times some integer. Instead of “n is divisible by d,” we can say that n is a multiple of d, or d is a factor of n, or d is a divisor of n, or d divides n. The notation d | n is read “d divides n.” Symbolically, if n and d are integers and d ≠ 0 :
Book notes
Course: Mathematics (MATH 271)
University: University of Calgary
