Math 230
F E Q(x) F Q(a) this rules for quantifiers means
that the variable x in the formula Q with a quantifier can be substituted with some specific a to make Q a false logic sentence
the logic switch representing p v q has two switches. which of the following statements is false about this circuit?
the switches are located one after the other
let p=>q, where p and q are sentences. then the following is true
then inverse and converse statements are equivalent
In the induction proofs how do we use P(k) and P(k+1)?
we assume P(k) is true and deducing P(k+1) is true
What do we do during an induction proof to show P(k+1) is true?
we assume P(k) is true and use our math knowledge and calculations to deduce P(k+1)
What is wrong with that proof? P : All tables are circular. STEP ONE: Choose a table that is round. P(1) is true. STEP TWO: We assume P(k) is true, for any set of k tables, they all round. Prove P(k+1) is true. Consider k+1 tables in a line. Take the first k tables. By the induction hypothesis they are all round. Now consider the last k tables. By the induction hypothesis they also are round. Since the sets overlap, we conclude that k+1 tables also are round. STEP THREE: Concluding P(n) is true hence, P: All tables are round.. WRONG!
we should have checked more than one table to star induction
what does not kill you makes you stronger. which sentences below are not equivalent to this sentence?
what can kill you makes you stronger
let A = {all 0,1 sequences starting with 1}. for example (1,1,1,1,0,0,0,0,0,....) is an element of A. the best way to establish the cardinality of A is to:
write the sequences in an increasing order???
R = {x is a real number} R = {y is a real number} R - numbers satisfy y^2 = x^2 For which numbers R is a binary relation?
x=0 and y=0
The graph of the parabola x=y^2 is a function X --- > Y only when:
y is positive ???
Which of the statements below give 1:1 correspondence between R= {reals} and the interval (0, Pi).
y=cot(x)
even * even = even and even +even = even odd * odd = odd ________________ odd * even = even Does the conclusion follow from the premisses?
yes, as we can represent odd = event + 1 and use the premises
U = {all students in our logic class}. Let A = { math majors}, B = {current calculus students}, C= {students who play an instrument } C n A' n B =
{ non math majors who take calculus students and play an instrument }
N = {natural numbers}. Check all correct statements about the power set 2N .
{0} is an element of 2N 2N contains the set of natural numbers N as an element 2N contains empty set as an element {1}, {1 ,2}, {1,2,3} , {1,2,3,4} are some of the elements of 2N 2N is infinite {even natural numbers} are an element of 2N
U = {all people in USA} . Let A = {all university students}, B = {people in California}. Then (A u B)' =
{Non Californians who are not students}
B - A: {P, O, N, T, R, E, L, I} - {G, I, A, N}
{P, O, T, R, E, L}
What are immediate consequences ~(X v Y) of the unsigned formula?
~X and ~Y
cut a round cake along the diagonal. cut a second time along a different diagonal. cut third time along a different diagonal. and so on. which statement P below is true?
after n cutes there will be 2n pieces
Consider these intervals as sets or real numbers (0, 1) , (-1, 0), (0, 5) . Which statement below is true about their cardinality?
all have the same cardinality aleph 1
you arrive on an isolated island where there are only two types of people - F they always lie, T always tell the truth. you ask each person: please tell me what types live on your island everyone answers 'some of us are type F and are good swimmers what can you conclude?
all of the same type, all are Fs and all do not swim
All students study hard. All who study hard pass the logic exam. _______________ Therefore, all students pass the logic exam. As we know the conclusion is not true, What went wrong with this argument?
both premises are false
degree of x => y = ??
degree of x => y = degree of x + degree of y + 1
completeness of tableaux method means that
every tautology is provable true by the tableaux method
X = N Y = N Which of the relations below are functions?
every x gets itself
let f and g be two functions well defined on real numbers, f: R -> R, g: R -> R which of the operations below are not well defined?
f/g
Which statement about finitely generated trees is true:
finitely generated trees may have infinitely many points
A a c A E b c B a<b means
for each element of a set there exists an element of the set B such that a < b
x,y,z c R X: A x c R E y c R E z c R 1 > Ix-yI > Ix-zI > 0 the statement above means
for every number x there exists a number y that is closer to it than one, and another number z that is even closer to z than y. both y,z are not equal to x
in the first logic predicates are
formulas with variables, quantifiers and connectives ~, v, =>, that can be decided to be true or false
to prove a statement P by induction we need to start with:
formulating P(n) and checking if it works for some small natural numbers
x - is a pet in X = {set of pets in US} properties: P - x is a good pet T - pet x gets a treat A x c X Tx => PX means:
if x gets a treat then x is a good pet (only good pets get a treat)
Well Ordering Principle helps to:
introduce an order on a subset of natural numbers from smallest to largest
Penrose calculated that there is a black hole in our galaxy. Earth circulate around the sun. _______________________________ Sun circulates around the black hole. Does this conclusion follow from the premisses?
no, both premises are true, but they are not connected
assume we need to prove that for some statement P: ∀n ∈N P(n) is even. We calculate and check: P(1) is odd. P(2) is even. P(3) is odd P(4) is odd. What is the next step if we need to use the proof by mathematical induction?
nothing can de done at this stage???
which of the following sentences are not an alternate denials?
only one of my friends passed the exam
PRINCIPLE OF FINITE DESCENT method helps to develop
processes/algorithms that will terminate
Principle of Mathematical Induction is a methodology to:
prove the statements about natural numbers
Let p and q be two sentences in logic. Consider the implication q => p Which of the following is true?
q => p is false when q is true and p is false
1+3+3^2+⋯+3^(n-1)=
(3^n −1 )/2
A = Z B = Z Let R be the following relation on A and B: a R b if a > b. Which of the pairs below are not in this relation? (5,7) (-5,-15) (500,0) (-7, -8) (-5,-7) (5,4) (0,-1) (1,0) (1000,-4)
(5,7)
Let Z be the universe for the following two sets A = { x > 4} and B = { 2 < x < 10}. What is (A n B)' = ?
(A n B)' = {. . . -1, 0, 1, 2, 3, 4, 10, 11. . . )
A = {Tom, Jim, Ann, Kim} B = {Tom, Jim, Ann, Kim} Tom, Jim, Ann are siblings, and Kim is Ann's best friend. Let R be the following relation on A and B: a R b if a is related to b. Which of the pairs below are not in this relation?
(Kim, Ann)
We want to match two sets of integers A = {integers} and B = {integers} using the following priorities. every a chooses b = a - 1 every b chooses a = b Using the method explained in class (Bernstin-Schroder theorem) which kind of matching into pairs (a,b) do we get?
(n, n-1)
let p and q be two sentences in logic consider the following (q v ~p) => q which of the following is true?
(q v ~p) => q is true when both q and p are true
Which number is on the following list: 0.1000000 ... 0.10100000.... 0.1010100000.... 0.10101010000.... 0.1010101010000.... and so on ...
0.10101010101010101010
A = {a, b, c}. Check the true statement about the power set of A (called 2 A )
2 A has cardinality 8
Z5 = {0,1,2,3,4} +,* In Z5, 11^2 * 23^2 =
4
degree of X = p v q v r => (r => q v r) is
5
Complete Induction Principle generalizes Induction principle in the following way: P is a statement about natural numbers. STEP ONE: Explore and formulate P. Find first several numbers for which P(n) is true. We can start our proof by complete induction. STEP TWO: We assume that for k < n and the statement P(k) is true. We need to show P(n) is true by using P(k) and our math knowledge. STEP THREE: Conclude P(n) is true for all n, hence Pis true.
???
Which of the examples below are examples of posteriori knowledge?
???
Amy has a set A = { all natural numbers divisible by 3} Ben has a set B = { all natural numbers divisible by 5} Chuck has a set C = { all natural numbers divisible by 15} They are comparing the sets. Which of their statements is NOT true A, B, C are infinite A, B , C have the same cardinality 3n ↔ 15n establishes 1:1 correspondence between A and C C is a subset of B A n B = C C is a subset of A A, B, C are subsets of natural numbers divisible by 30 A, B, C are subsets of natural numbers A, B , C are countable
A, B, C are subsets of natural numbers divisible by 30
Which operations on numbers are well defined in the set of natural numbers N?
Addition and multiplication only
Which operations on numbers are well defined in the set of integers Z?
Addition, subtraction and multiplication only
Which operations on numbers are well defined in the set of rational numbers Q?
Addition, subtraction, multiplication and division (except by 0) only
Which operations are well defined in the set of complex numbers C?
Addition, subtraction, multiplication, division (except by 0) and solving algebraic equations
Which operations on numbers are well defined in the set of real numbers R?
Addition, subtraction, multiplication, division (except by 0) and taking radicals and limits
which of the following sentences is a joint denial?
Amy is neither a good athlete nor a good musician
A is a subset of B. Which of the following is NOT true?
B - A = empty set
C = { natural numbers divisible by 3}. Which statement about C is not rue? 3n is a formula for elements of C 3n < --- > n is 1:1 correspondence with natural numbers C ={ 3,6,9,12,15,18 ..... } numbers divisible by 6 belong to C C is an infinite set adding two elements of C give another element of C C is uncountable 270 belongs to C C is countable all these statements are true
C is uncountable
THE LEAST NUMBER PRINCIPLE says that:
Every non-empty subset of N contains a smallest element.
Which are immediate conclusion of the signed formula F X v Y?
FX and FY
When two infinite sets are equal?
If all their elements are the same
If all elements of an infinite set A are natural numbers then there exists the smallest element in this set. Which of the following is equivalent to the statement above?
If there is no the smallest element in this set, then not all elements of a set A are natural numbers
R (the set of reals) is the universe . A= (1,4 ) B= [-3, 2] R+ = {all positive decimals} . Check all true statements below.
R+ - B = (2, infinity) R+ u (R+ )' = R A n B = (1, 2] A - R+ = Empty set A u B = [-3, 4)
Which is a true statement in logic?
There are statements that are always true
Which is a false statement about axioms in mathematics?
We do not need to state the axioms to use them
When an argument is valid?
When premises are true, then the conclusion must be true
When two infinite sets have the same cardinality?
When there is 1:1 correspondence between their elements
A x X(x) => X(a) means that
X is true for all x, so in particular X is true for specific x=a
M= {all finite (strings) words that can be made from letters in A}. What is the cardinality of M? Justify.
X1 so countable infinite; therefore, 1:1 correspondence with N
read the following true statement: for every natural numbers n+m=m+n. how this knowledge was acquired by us?
a priori - it can be found through reason alone
Z6 = {0,1,2,3,4,5} +,* which of the following is not true in Z6
a*b=0 means a or b has to be equal to zero
the implication p=>q is false
only if the sentence p is true and the q is false
Formula X is equivalent to y means
that both are true or both are false for all logical values of their sentences
A - B: {G, I, A, N} - {P, O, N, T, R, E, L, I}
{G,A}
Let P:n^2 - n - 4 > 0 check which P(1), P(2), P(3), P(4), P(5) are true
P(3), P(4), P(5) are true
we start with the 1 x 1 square. then we double each side to obtain a new square. then we double the side again to obtain even larger square. and so on. which is a correct statement P about the area A of the n-th square?
A = (2^n)(2^n)
A= { x, y, z } FS = {finite sequences using letters x, y, z } , such as xxyxyzz IS = {infinite sequences using letters x, y, z } , such as xxyxyzzxzy .... Which of the statements below are true? neither of these answers is true A is infinite, FS is countable, IS is uncountable IS is finite FS and IS have the same cardinality elements of FS are finite, hence FS is finite FS and IS are identical sets A is finite, FS is countable, IS is uncountable A is finite, FS is uncountable, IS is countable FS is empty FS finite, A is finite A empty, FS is countable, IS is countable only finite sequences are in IS
A is finite, FS is countable, IS is uncountable
Which is the best 'definition' of the term 'definition' in mathematics?
A statement of the exact meaning of a concept or precise description of an object
For any two sets ( C u B)' =
C' n B'
To prove that fractions are countable we need to make list of them. Which is not true about any enumeration proof? Just putting fraction into a matrix is enough Adding few extra fractions to the list does not change the argument We should skip repetitions Numbers on diagonals have the same sum of denominator and numerator There is a way to figure out the formula for this sequence The list should clarify what is a position of each fraction on the list We can concentrate on positive fractions first Cardinality of fractions is smaller than cardinality of real numbers
Just putting fraction into a matrix is enough
L= { all lines y=ax+b on a plane such that a, b are both integers} Which statement about L is NOT true? L is infinite Lines y=ax form a subset of L L has as many elements as N L is a subset of P= { all polynomials} Each line corresponds to a point (a,b) L is not countable y=2x+2 is and element of L L is countable There as as many lines in L as positive fractions
L is not countable
Which of the following are valid for any sets F and G: 1) F - G = G - F 2) F' = G 3) F u G = G' u F' 4) F u G = G u F 5) F n G = G n F 6) F n G - F = G - G
Last 3 are valid
Consider R to be the universe for the following sets. N = (1,2,3,4 .... ) Z= {...-3, -2, -1, 0, 1, 2 , 3 ....} Even = ( ...-4, -2, 0, 2 , 4 ...} R+ = {all positive decimals} Note : u - means 'union'; n means 'intersection' , A' - means a complement to A Check all statements below that are true.
N u R+ = R+ Z - Even = Odd Z u Even = Z (R+ )' = {all non positive reals} (Z n N )'= {all reals that are not natural numbers} R+ - Z = {all positive real numbers that are not integer}
p: I win a lottery q: I give you a million dollars I promised you: if I win a lottery I will give you a million dollars. In which case I lied to you?
Only in case I win a lottery, but I do not give you a million dollars
Let P = { quadratic polynomials that can be factored as (x-a)(x-b) where both a and b are integers} Which statement about P is true? Cardinality of P is aleph 1 a has to be different than b a and b have to be positive P is not countable P is a finite set P is a subset of Z ={integers} P is countable P has a subset of not factorable polynomials All polynomials are factorable in real numbers
P is countable
what can we conclude about the statement P after we are done with the induction step?
P is true for all natural numbers bigger or equal to the smallest number for which P(n) is true
Consider the set S = {all sequences with terms that are digits from 0 to 9} . For example 1,2,3,3,3,2,5,6 ... is an element of S. Which statement is not true? Sequences in S do not have to converge S is not countable S is countable 1,1,1,1,1,1,1,...........is and element of S There is 1:1 correspondence with decimals from the interval [0,1) S has cardinality aleph 1 There are as many sequences in S as there are non negative decimal S is a subset of T = {all sequences with terms are integers} .
S is countable
Let F = {all positive fractions smaller than 1}. Which of the following statements is not true : F is a subset of Q We can number the elements of F 1,2,3,4 ... We can make a list of the elements of F F has as many elements as Q F is a subset of the interval (0, 1) The cardinality of F is aleph 0 F is countable Some of these statements are false F has as many numbers as N does F can be enumerated
Some of these statements are false
Is Z = [p v q] => p tautology? FZ = F[p v q] => p which is the next step in the proof by tableaux method?
T[p v q] Fp
Let S= {all (strings) words that can be made from letters in A that are shorter than 5 letters}.
The cardinality is 340 if there are 4 letters (4^1 + 4^2 + 4^3 + 4^4)
Which of the following statements is not true about the power sets? For finite sets power sets are finite Empty Set is an element of any power set For infinite sets power sets are infinite Any set A is an element of its power set For countable sets the power sets are not coutable The cardinality of the power set for a given non empty set A can be equal to the cardinality of A Power sets are used to construct larger cardinal numbers
The cardinality of the power set for a given non empty set A can be equal to the cardinality of A
Mark all sets that have the set of rational numbers Q as a proper subset.
The set of complex C and the set of real numbers R
Mark the sets that are NOT contained in the set of real numbers R as a proper subset.
The set of imaginary numbers