EXam one Math 3000

The Cartesian product

of two sets A and B is another set, denoted as AxB and defined as AxB= {(a,b): a∈A, b∈B}

A non-constructive proof of an existence statement proves an example exists without actually giving it .

proves an example exists without actually giving it

what explains the process of proving a conditional statement "If P, then Q" using a contrapositive proof?

Suppose that ~Q. . . . Therefore, ~P.

Suppose set 𝐴={2,4,8} and set 𝐵={3,9,27,81} Which of the following elements would be in the set 𝐴×𝐵A×B? Select ALL elements in 𝐴×𝐵A×B if more than one of the options below is in 𝐴×𝐵A×B.

(4,9) and (2,81)

You wish to prove the following: Proposition: For every positive integer n, If you are going to prove this by induction, what is the initial value of n we need to check for the Base Case? Enter the value for the base case of n in the blank.

1

Suppose set 𝐴={2,4,8}A={2,4,8} and set 𝐵={3,9,27,81}B={3,9,27,81}. How many elements will be in the set 𝐵×A?

12

Suppose A = { 0 , 1 , 2 , 3 , 4 } and B = { 5 , 10 , 15 , 20 , 25 , 30 , 35 }, and let define the relation f = { ( 0 , 15 ) , ( 1 , 20 ) , ( 2 , 5 ) , ( 3 , 15 ) , ( 4 , 35 ) }. What is the DOMAIN of f?

A = { 0 , 1 , 2 , 3 , 4 }

definition of the partition function

A Partition of a set A is a set of non-empty subsets of A, such that the union of all the subsets equals A, and the intersection of any two different subsets is the empty set

Prime number

A number n ∈ N is prime if it has exactly two positive divisors, 1 and n. If n has more than two positive divisors, it is called composite. (Thus n is composite if and only if n = ab for 1 < a,b < n.)

Perfect number

A number p ∈ N is perfect if it equals the sum of its positive divisors less than itself. (ex) The number 6 is perfect since 6 = 1+2+3.

Rational/irrational number

A real number x is rational if x = a /b for some a, b ∈ Z. Also, x is irrational if it is not rational, that is if x not= a/b for every a,b ∈ Z.

Definition of an equivalence relation

A relation R on a set A is an equivalence relation it it is reflexive, symmetric and transitive

Definition of a relation

A relation on a set A is a subset R⊆AxA. We often abbreviate the statement (x,y) e R as xRy. The statement (x,y) not (e) R is abbreviated x notR y

even integer

An integer n is even if n = 2a for some integer a ∈ Z.

odd integer

An integer n is odd if n = 2a+1 for some integer a ∈ Z

Suppose A = { 0 , 1 , 2 , 3 , 4 } and B = { 5 , 10 , 15 , 20 , 25 , 30 , 35 }, and let define the relation f = { ( 0 , 15 ) , ( 1 , 20 ) , ( 2 , 5 ) , ( 3 , 15 ) , ( 4 , 35 ) }. What is the CODOMAIN of f?

B = { 5 , 10 , 15 , 20 , 25 , 30 , 35 }

When proving a conditional statement such as "If P, then Q.", we can first identify what are statements P and Q. Then, if we are going to use direct proof, which ONE of the options below would be a reasonable starting point?

First, we assume P is true. Then we try to show that this forces Q to be true as well.

Domain and codomain

For a function f : A → B, the set A is called the domain of f . (Think of the domain as the set of possible "input values" for f .) The set B is called the codomain of f . The range of f is the set {f (a) : a ∈ A } = { b : (a,b) ∈ f } . (Think of the range as the set of all possible "output values" for f . Think of the codomain as a sort of "target" for the outputs.)

what gives the negation of the following statement? There exists an integer a such that 2a+1 is even.

For all integers a, it follows that 2a+1 is odd.

You want prove the following. For all positive integers n, statement Sn is true. Which of the following conditional statements can be used to prove the induction step?

For an integer 𝑛≥n≥1, assume that 𝑆𝑛 is true . Then we want to show that 𝑆_(n+1) is true. For an integer 𝑘≥k≥1, assume that 𝑆𝑘 is true . Then we want to show that 𝑆𝑘+1 is true. Assume that 𝑆𝑘 is true for all 1≤𝑘<𝑛≤k<n. Then we want to show that 𝑆𝑛 is true for all integers 𝑛>k.

We want to prove the following statement using proof by induction: If 𝑋={1,...,𝑛}, then |P(X)|=2^n When we prove the induction step, which of the following correctly states the induction hypothesis that statement P(k) is true for all integers k in the interval 𝑏≤𝑘<𝑛b≤k<n, where 𝑏b denotes the base case value of the integer. Which of the following correctly gives the induction hypothesis for this example?

For k an integer in the interval 1≤𝑘<𝑛1≤k<n, we assume that if 𝑋={1,...,𝑛}, then |P(X)|=2^k

Question 6 from Worksheet 13: We want to prove the following statement using proof by induction: If 𝑋={1,...,𝑛}, then |P(X)|=2^n In the inductive step, we assume the induction hypothesis is true. Which of the statements below correctly states what we need to prove in the inductive step?

For 𝑛>𝑘n>k, we want to show that if 𝑋={1,...,𝑛}, then |P(X)|=2^n

Congruent modulo

Given integers a and b and n ∈ N, we say that a and b are congruent modulo n if n | (a− b). We express this as a ≡ b (mod n). If a and b are not congruent modulo n, we write this as a 6 not≡ b (mod n). (ex) 9 ≡ 1 (mod 4) because 4 | (9−1)

Definition of Power Set

If A is a set, the power set of A is another set, denoted as P(A) and defined to be the set of all subsets of A. In symbols, P(A) = { X : X ⊆ A }.

Open sentences

If P and Q are statements or open sentences, then "If P, then Q," is a statement. This statement is true if it's impossible for P to be true while Q is false. It is false if there is at least one instance in which P is true but Q is false.

Converse of "If P, then Q." is equivalent to

If Q then P

Let A = { x : P ( x ) }. If we want to show that A ⊆ B, then we need to show that:

If a ∈ A, then a ∈ B.

We want to prove the following statement using proof by induction: If 𝑋={1,...,𝑛}, then |P(X)|= 2^n Which of the following is a correct proof for the Base Case?

If n=1 , then X= {1}. Thus, we have P(x)= {0, {1}} and |P(x)|=2. So when n=1, we see that |P(x)|=2^1=2

Contrapositive of "If P, then Q." is equivalent to

If not Q, then not P

You wish to prove the following: Proposition: For every positive integer n, If you are going to prove this by induction, which of the following is the most appropriate proof for the base case? Select one.

If 𝑛=1, then 1=1⋅221=1⋅22, which is true.

A sequence (𝑎𝑛)∞𝑛=1(an)n=1∞ is defined recursively by 𝑎1=1,𝑎2=3, and 𝑎𝑛=2𝑎𝑛−1−𝑎𝑛−2 for 𝑛≥3.a1=1,a2=3 ,and an=2an−1−an−2 for n≥3. Prove that 𝑎𝑛=2𝑛−1an=2n−1 for all 𝑛∈ℕn∈N. At the end of last class, we proved the base cases and set up the induction hypothesis (see annotated notes from last class). Below as a sample of a possible completed proof. Induction Step: For integers 2≤𝑘<𝑛2≤k<n, assume that 𝑎𝑘=2𝑘−1.ak=2k−1. We want to show that for integers 𝑛>𝑘n>k we have 𝑎𝑛=2𝑛−1an=2n−1. Consider an integer 𝑛>𝑘n>k. By inductive hypothesis, we know: 𝑎𝑛=2𝑛−1=(4𝑛−6)−(2𝑛−5)=2(2𝑛−3))−((2𝑛−4)−1)=2(2(𝑛−1)−1)−(2(𝑛−2)−1)=2𝑎𝑛−1−𝑎𝑛−2.an=2n−1=(4n−6)−(2n−5)=2(2n−3))−((2n−4)−1)=2(2(n−1)−1)−(2(n−2)−1)=2an−1−an−2. Thus we can see the property hold for all positive integers. Decide whether or not the induction step above is correct. Pick the one answer that best answers this question.

It is not correct because this person has assumed what they are being asked to prove.

You wish to prove the following: Proposition: For every positive integer n, If you are going to prove this by induction, which of the following is the most appropriate start for a proof of the induction step? Select one.

Let 𝑛>1 and assume that for any 𝑘k in 1≤𝑘<𝑛1≤k<n it follows that the property Let 𝑛>n>1 and assume that for any 𝑘k in 1≤𝑘<𝑛1≤k<n it follows that the property 1+2+3+...+𝑘=𝑘(𝑘+1)/2...1+2+3+...+k=k(k+1)2 is true.

Which of the symbols below is used to denote the natural numbers? Note the natural numbers are { 1 , 2 , 3 , 4 , ... }.

N

Let A = { x : P ( x ) }. If we want to show that a ∈ { x : P ( x ) }, then we need to show that:

P ( a ) is true.

Indexed sets definition

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Which of the following gives an appropriate way to disprove a conditional statement of the form "If P ( x ), then Q ( x )."?

Produce an example of an x that makes P ( x ) true and Q ( x ) false.

Which of the following gives an appropriate way to disprove a universally quantified statement of the form ∀ x ∈ S , P ( x )?

Produce an example of an x ∈ A that makes P ( x ) false.

Which of the symbols below is used to denote the rational numbers (numbers that can be written as p/q for some integers p and q)?

Q

Which of the symbols below is used to denote the real numbers?

R

Reflexive symmetric transitive

Relation R is reflexive if xRx for every xeA. Relation R is symmetric if xRy implies yRx, then also xRz Relation R is transitive if whenever xRy and yRz, then also xRz

Let A = { x : P ( x ) }. If we want to show that A = B, then we need to show that:

Show thatA ⊆ B and B ⊆ A.

Prove that (root7) is irrational. If you are going to prove the statement above using Proof by Contradiction, which of the sentences below could be the first sentence of your proof by contradiction?

Suppose (root7) is rational.

set with functions

Suppose A and B are sets. A function f from A to B (denoted as f : A → B) is a relation f ⊆ A × B from A to B, satisfying the property that for each a ∈ A the relation f contains exactly one ordered pair of form (a,b). The statement (a,b) ∈ f is abbreviated f (a) = b.

Definition of function

Suppose A and B are sets. A function f from A to B is a relation

Definition of Subsets

Suppose A and B are sets. If every element of A is also an element of B, then we say A is a subset of B, and we denote this as A ⊆ B. We write A not⊆ B if A is not a subset of B, that is, if it is not true that every element of A is also an element of B. Thus A not⊆ B means that there is at least one element of A that is not an element of B.

Equivalence class relating definition

Suppose R is an equivalence relation on a set A. Given any element aEA, the equivalence class containing a is the subset {xeA :xRa} of A consisting of all the elements of A that relate to a. This set is denoted as [a]. Thus the equivalence class containing a is the set [a] ={xeA :xRa}

Divides

Suppose a and b are integers. We say that a divides b, written a | b, if b = ac for some c ∈ Z. In this case we also say that a is a divisor of b, and that b is a multiple of a.

Let a be an integer. If a^2 is odd, then a is odd. If you are going to prove the statement above using Proof by Contradiction, which of the sentences below could be the first sentence of your proof by contradiction?

Suppose a^2 is odd and a is even.

Select ALL of the statements below that are TRUE. Be sure to select multiple answers if you believe more than one is correct. You may assume A and 𝐵 denote sets.

The cardinality of 𝐴×B is equal to the cardinality of A times the cardinality of B.

Definition of a complement

The complement of A, denoted Abar is the set of Abar= U-A

Consider the following statements. P is "Colorado has mountains." Q is "Colorado borders an ocean." Hint: Colorado does indeed have mountains and there are no ocean beaches in Colorado. In this case, is the conjunction P ∧ Q true or false?

The conjunction P ∧ Q is a FALSE statement.

Consider the following statements. P is "Colorado has mountains." Q is "Colorado borders an ocean." Hint: Colorado does indeed have mountains and there are no ocean beaches in Colorado. In this case, is the disjunction P ∨ Q true or false?

The disjunction P ∨ Q is a TRUE statement.

We want to prove a statement of the following form: For all integers 𝑛≥𝑏n≥b, statement 𝑃(𝑛)P(n) is true. Mathematical induction in general can informally summarized as follows: (1) Base case: Show the statement is true for the base case when 𝑛=𝑏n=b. (2) Inductive step: If we assume the statement P(k) is true for "smaller" integers 𝑘k in the interval 𝑏≤𝑘<𝑛b≤k<n, Then we can show the statement P(n) is true for "larger" integers 𝑛>𝑘n>k. (3) Thus the statement P(n) is true for all integers 𝑛≥𝑏n≥b We've been using a chain of dominos knocking each other over as an analogy for this process. Match each of the steps of proof by induction to the its corresponding step in the domino analogy.

The first domino falls down. (prove the base case p(b) is true) The first k dominos are all knocked down by a previous domino. (Assume the inductive hypothesis is true) Then the dominos after the kth domino will also be knocked over. (Prove p(n) is true for n>k.

Greatest common divisor

The greatest common divisor of integers a and b, denoted gcd(a,b), is the largest integer that divides both a and b. The least common multiple of non-zero integers a and b, denoted lcm(a,b), is the smallest integer in N that is a multiple of both a and b

what gives the negation of the following statement? For every prime number p, the number p+1 is not prime.

There exists a prime number p such that p+1 is also prime.

Consider the following sentence: "The integer x is a positive number." Determine whether or not the sentence above is a statement. Which one of the answer choices best explains why the sentence is or is not a statement?

This is NOT a statement since the sentence is sometimes true and sometimes false. Thus, this is an OPEN SENTENCE.

Consider the following sentence: "Go brush your teeth." Determine whether or not the sentence above is a statement. Which one of the answer choices best explains why the sentence is or is not a statement?

This is NOT a statement since this is a command which is neither true nor false.

Consider the following sentence: "Today the sun is shining, and the weather is warm" Determine whether or not the sentence above is a statement. Which one of the answer choices best explains why the sentence is or is not a statement?

This is a statement since although we cannot tell whether it is true or false, we can be sure that the sentence either must be true or it must be false.

Consider the following sentence: "7 is an even number." Determine whether or not the sentence above is a statement. Which one of the answer choices best explains why the sentence is or is not a statement?

This is a statement since we can determine the sentence is false.

Consider the following sentence: "If x is an integer, then 2x is an even number." Determine whether or not the sentence above is a statement. Which one of the answer choices best explains why the sentence is or is not a statement?

This is a statement since we can determine the sentence is true.

Two equal functions

Two functions f : A → B and g : A → D are equal if f = g (as sets). Equivalently, f = g if and only if f (x) = g(x) for every x ∈ A.

Parity

Two integers have the same parity if they are both even or they are both odd. Otherwise they have opposite parity.

Union, intersection and difference definitions

Union: AUB= {x:x∈ A or x∈B} Intersection: AUB= {x: x∈A and x∈B} Difference: A-B = {x: x∈A and x not∈B}

Imagine we want to prove that every postage amount n of at least 8 cents can be made exactly with stamps of 3 cents and 5 cents. In proving the base case, what is the minimum number of base cases we need to check before moving on to the induction step?

We need to check the cases n=8, n=9, and n=10.

Suppose A = { 0 , 1 , 2 , 3 , 4 } and B = { 5 , 10 , 15 , 20 , 25 , 30 , 35 }, and let define the relation f = { ( 0 , 15 ) , ( 1 , 20 ) , ( 2 , 5 ) , ( 3 , 15 ) , ( 4 , 35 ) }. Is this relation a function? Explain.

Yes, since for each element in set A is paired with exactly one element in B.

Which of the symbols below is used to denote the integers? Note the integers are { 0 , ± 1 , ± 2 , ± 3 , ... }.

Z

Definition of a relation

a relation from a set A to a set B is a subset R⊆AxB. Abbreviate the statement

Which of the following functions are one-to-one (also called injective)? aka they pass the horizontal line test

f ( x ) = 1 /x f ( x ) = 5 x − 7 f ( x ) = e x

To prove an existence statement, we need to show that a universally quantified statement is false. .

give one example that shows it is true,

Injective surjective bijective

injective (or one-to-one) if for all a, a' ∈ A, a not= a' implies f (a) not= f (a') surjective (or onto B) if for every b ∈ B there is an a ∈ A with f (a) = b; bijective if f is both injective and surjective.

Definition of an ordered pair

is a list (x, y) of two things x and y, enclosed in parenthesis and separated by a comma

Which of the symbols below is used to denote "in"?

∈

definition of integers modulo n

let n eN. The equivalence classes of the equivalence relation= (mod n) are [0],[1],[2].....[n-1]. The integers modulo n is the set Z_n= {[0],[1],[2],...[n-1]}. Elements of Z_n can be added by the rule [a] +[b]= [a+b] and multiplied by the rule [a]*[b]= [ab]

Fermats last therom

that no three positive integers a, b, and c satisfy the equation a^n + b^n = c^n for any integer value of n greater than 2.

Cardinality

the number of elements in a set or other grouping, as a property of that grouping.

Suppose A = { 0 , 1 , 2 , 3 , 4 } andB = { 5 , 10 , 15 , 20 , 25 , 30 , 35 }, and let define the relation f = { ( 0 , 15 ) , ( 1 , 20 ) , ( 2 , 5 ) , ( 3 , 15 ) , ( 4 , 35 ) }. What is the RANGE?

{ 5 , 15 , 20 , 35 }

Let A = { 3 , 5 , 7 }. Which of the following is the power set of A, which we denote with P ( A )?

{ ∅ , { 3 } , { 5 } , { 7 } , { 3 , 5 } , { 3 , 7 } , { 5 , 7 } , { 3 , 5 , 7 } }

Which of the symbols below is used to denote "for all"?

∀

Which of the symbols below is used to denote "there exists"?

∃

Consider the following statements. P is "Colorado has mountains." Q is "Colorado borders an ocean." Hint: Colorado does indeed have mountains and there are no ocean beaches in Colorado. In this case, is ∼ Q true or false? Recall ∼ Q denotes the negation of Q.

∼ Q is a TRUE statement.

Which of the symbols below is used to denote a conditional "if ... then ..."?

⟹

Which of the symbols below is used to denote a biconditional "... if and only if ..."?

⟺

Which of the following properties must be true in order for 𝑅R to define an equivalence relation on A? Select ALL that apply if more than one answer is correct. Note that 𝑥,𝑦,𝑧∈𝐴

𝑥𝑅𝑥 for all 𝑥∈𝐴 If 𝑥𝑅𝑦, then it follows 𝑦𝑅𝑥 If 𝑥𝑅𝑦 and 𝑦𝑅𝑧, then it follows 𝑥𝑅𝑧