Tenta Matematik och logik

Find the gcd(420,66) by using the Euclidean algorithm

svar: 6

What does this represent? A ⊆ B

A is a subsett of B

Vad kommer settet A innehålla?

A = {0,1, 4}

Power set

Power set

What is the cardinality of a set?

total number of unique elements in a set A = {1,2,3}, cardinality of A = 3

What is the prime factorisation of 455?

5 * 7 * 13

T(1) = 1 T(n) = T(n-1) + 3 Vad blir T(1),T(2)(, T(3), T(4), T(5)

T(1) = 1 T(2) = 4 T(3) = 7 T(4) = 10 T(5) = 13

Klicka för svar

n = 1: 1 n = 2: 1+1 n = 3: 1+1+1, so we see that the output is n

How many five-card poker hands can be dealt from a standard 52-carddeck?

n = 52 r = 5 52! / (52-5)! * 5! 311 875 200 / 5! = 2 598 960 2 598 960 different ways

Find the coefficient of x^3 in expansion of (2x + 1)^4

nCr (a)^n-r * (b)^r Vi vet att x = x^3 och a^n-r + (b)^r måste vara lika som n då blir b (b)^1 r = 1

let n be N n^2 =! n give one counter example

svar: n = 1 n^2 = n

What is ϕ(2)?

svar: 1 (the number 1)

What does this symbol mean? M ⊂ S

if M ⊆ S and M ≠ S, then there is at least one element of S that is not an element of M, then M is a proper subset of S.

The set S = {1, 2, 3} and T = {2, 4, 7} What is the relation p when x p y <--> x = y/2

p = {(1,2),(2,4)}

S = {2, 4, 6, 8} and T = {2, 3, 4, 6, 7} What is the set p that satifies the realtion x r y <--> x = (y + 2)/2?

p = {(2,2),(4,6)}

When are two sets equal?

only if they contain the same elements.

Prove that the sum of 5 consecutive integers is divisible by 5.

x-2, x-1, x, x+1, x+2 Sum, S = x-2+x-1+x+x+1+x+2 S = 5x

What is the symbol of a superset?

B⊇A B is a superset of A

Every fridge contains milk

F (x): x is a fridge, M (x): x contains milk

F(1) = 1 F(2) = 1 F(n) = F(n-1) + F(n-2) for n >= 2 Vad blir F(1), F(2), F(3), F(4), F(5), F(6), F(7)

F(1) = 1 F(2) = 1 F(3) = 2 F(4) = 3 F(5) = 5 F(6) = 8 F(7) = 13

Vad är en Transitive relation?

If you have two different arrows, then you must have a third arrow if you have an arrow from (x,y) and (y,z) you must have an arrow to (x,z) a --> c --> d, då måste det finnas en pil mellan a --> d

What can we conclude if you do get strong?

We can conclude that you will win races

What is the prime factorisation of 90?

90 = 2 x 3 x 3 x 5

What does the following function calculate?

What does the following function calculate?

Förklara Euler's Phi Function

https://www.youtube.com/watch?v=XDnksOa3GMk Dom måste vara relativly prime, alltså den enda gcden måste vara 1

Find the 4th term of (3x + 2y)^5

https://www.youtube.com/watch?v=Y32hGegCoPI

What is the fifth term of (3a + 2b)^7

https://www.youtube.com/watch?v=Y32hGegCoPI

Find the coefficient of x^12 in expansion (2x^3 - 2)^5

https://www.youtube.com/watch?v=gdmKkRYdiyA

What is ϕ(6)?

Svar: 2 (the numbers 1,5)

What is ϕ(5)?

Svar: 4(the numbers 1,2,3,4) Om det är ett primtal så blir gcd alltid 1

Convert the following verbal arguments into propositional logic. Remember to first define your variables. "If you train hard and eat good food, then you will get strong. If you get strong, then you will win races."

T= I train hard E = I eat good food S = I am strong W = I win races svar: (T Λ E) → S, S → W

what does S ⊃ M mean?

This means that S is a proper superset of M if M is a proper subset of S, then S is a proper superset of M.

What does this symbol mean? |S|

This means the cardinality of S The cardinality of a set is the number of unique elements in a set

What does this symbol represent? Ø

This symbol represent a empty set which contains no elements

Vad är en symmetric relation?

Tänk att du har relationen (x,y) och (y,x) Om du har en pil från x till y, då måste du också ha en pil från y till x.

What is the cartesian product of A and B A= {a,b,c} B={1,2,3}

A x B = {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3), (c, 1), (c, 2), (c, 3)}

If A = { 1,3,5,7,9} and B = {3,7,9,10,15} What is: A ∩ B A ∪ B

A ∩ B = {3,7,9} A ∪ B = {1,3,5,7,9,10,15}

prove by contraposition: if n is an integer and 3n + 2 is even, then n is even

Assume n is odd. so n = 2k+1 3n +2 = 3(2k + 1) + 2 = 6k + 3 + 2 = 2(3k + 2) + 1 = 2r + 1 where r = 3k + 2 this shows that 3n + 2 is odd. We have proved q' --> p', then by contraposition p --> q is true

Let x, y be R (x > y) → (x^2 > y^2) give one counter example

svar: x = 5 y = -10 x^2 = 25 y^2 = 100 25 < 100

What is ϕ(4)?

svar: 2 (the numbers 1,3)

What is ϕ(3)?

svar: 2 (then numbers 1,2)

What is ϕ(7)?

svar: 6 (the numbers 1,2,3,4,5,6) Om det är ett primtal så blir gcd alltid 1

825 = 3 x 5 x 5 x 11 455 = 5 x 7 x 13 What is the gcd of (825,455)

svar: gcd(825,455) = 5

When are two integers a relatively prime?

svar: if gcd(a,b) = 1

Let x, y be R (x > 0) → (7x^3 > x^4) give one counter example

svar: x = 10 7*10^3 = 700010^4 = 100007000 < 10000

Let A = {1, 2, 3, 5, 10} B = {2, 4, 7, 8, 9} C = {5, 8, 10} be subsets of S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Find : A ∪ B

svar: {1, 2, 3, 4, 5, 7, 8, 9, 10}

Let A = {1, 2, 3, 5, 10} B = {2, 4, 7, 8, 9} C = {5, 8, 10} be subsets of S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Find : A - C

svar: {1, 2, 3} subtract the elements from set A witch are similar to C

Let A = {1, 2, 3, 5, 10} B = {2, 4, 7, 8, 9} C = {5, 8, 10} be subsets of S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Find : B' ∩ (A ∪ C)

svar: {1,3,5,10}

Let A = {1, 2, 3, 5, 10} B = {2, 4, 7, 8, 9} C = {5, 8, 10} be subsets of S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Find : A ∩ B ∩ C

svar: Ø

skippa denna fråga är inte klar A: b|a B: a > 0 C: b ≤ a A ∧ B → C Prove by contradiction that S is true

the negation of this A ∧ B → C is this A∧B∧C′ C'. then means b > a B. implies a > 0, A. b|a we also know that there exists an integer n such that a = n *b beacuse a > 0, then nb > 0 vilket leder till att n > 0 om utgår från C' = b > a och b är större än nb det betyder att för att b ska vara större, då måste n vara <= 0 då har vi alltså n > 0 & n < 0 vilket är en contradiction Därför vet bi att C är true eftersom C' inte är true

In how many ways can three athletes be declared winners from a group of 10 athletes who compete in an Olympic event?

n = 10 r = 3 C(10,3) = 120

What are real numbers? how are they represented?

real number, in mathematics, a quantity that can be expressed as an infinite decimal expansion. tex pi eller e, they are represented by R It is also integers (-2, 0, 1), fractions(1/2, 2.5) and irrational numbers such as √3

What is the prime factorisation of 825?

3 * 5 * 5 * 11

Vad betyder detta? b∉A

Detta betyder att element b inte finns i sett A

Hur kommer listan att bli sorterad?

1. 23, 12, 9, -3, 54, 89 2. 23, 12, 9, -3, 54, 89 3. -3, 12, 9, 23, 54, 89 4. -3, 9, 12, 23, 54, 89 5. -3, 9, 12, 23, 54, 89

A = {x | x ∈ N and x >= 5} --> {5,6,7,8,9,.......} B = {10,12,16,20} C = {x | (∃y)(y ∈ N and x = 2y)} --> {0, 2, 4, 6, 8, 10, .......} which alternativs are true? 1. B ⊆ C 2. A ⊇ C 3. {11,12,13} ⊆ A 4. {12} ∈ B 5. {Ø} ⊆ B 6. B ⊂ A 7. 26 ∈ C 8 {11,12,13} ⊂ C 9. {12} ⊆ B 10. 5 ⊆ A 11. Ø ⊆ A

1. sant eftersom 10,12,16 och 20 också finns i settet C 3. Detta är sant eftersom 11,12,13 finns i settet A 6. Detta är sant eftersom alla siffror som finns i B finns också i A. men alla siffror i A finns inte i B, alltså B är ett proper subset till A. 7. 26 finns i C, ja det gör den 9. 12 är ett subset av B 11. true, since Ø contains no elements, all the elements in Ø must be in A. therefore Ø is a subset of A

From a group of 10 men and 12 women, how many committees of 5 men and 6 woman can be formed?

10C5 = 252 different groups of men can be formed 12C6 = 924 groups of women can be formed 252 * 924 = 232,848

Find gcd( 544, 212) using the ecludian algorithm

544 = 212(2) + 120 gcd(212,120) 212 = 120(1) + 92 gcd(120,92) 120 = 92(1) + 28 gcd(92,28) 92 = 28(3) + 8 gcd(28,8) 28 = 8(3) + 4 gcd(8,4) 8 = 4(2) + 0 svar: gcd = 4 https://www.youtube.com/watch?v=5LhWIaWlprM&t=196s

Find gcd(414, 662) using the ecludian algorithm

662 = 414(1) + 248 414 = 248(1) + 166 248 = 166(1) + 82 166 = 82(2) + 2 82 = 2(41) + 0 svar: gcd = 2

What is a prime number? Give examples of prime numbers

A number is prime if the only numbers that divide it leaving no remainders are itself and 1 2,3,5,7,11

When is a set a subset?

A set is a subset of another if all its elements are also in the other set. For example if A = { 2,4,6} and B = {1,2,3,4,5,6} Then A is a subsett of B.

What is a superset?

A superset is the opposite of subset. If M is a subset of S, then S is a superset of M

The bull's-eye is the innermost circle on a dartboard. Use propositional logicand logical connectives to prove that if a dart has not hit the board, then itcannot have hit the bull's-eye. Remember to first define your statement letters.

A: dart hits the bull's-ey B: dart hits the board A': dart has not hit the bull's-ey B': dart has not hit the board the bull's-ey is part of the board, so A --> B this is true Proof by contraposition: In contraposition if p --> q is true, then it is equivilant to q' --> p' By contraposition we can say thath B' --> A'

if n is an integer and 3n + 2 is odd, then n is odd prove this by contraposition

Assume n is even. So n = 2x 3n + 2 = 3(2x) + 2 = 6x + 2 = 2(3x+1) = 2 y where y = 3x + 1 This shows that 3n + 2 is even. Since q' --> p' is true, then p --> q is true

What is a number that is not a prime number a product from?

Any number that is not a prime number is a product of prime numbers. for example 12 = 3 x 2 x 2 This number is called a composit number

If a bike has no wheels, then it cannot be ridden

B(x) = x is a bike W(x) = x has wheels R(x) ) x can be ridden ∀x(B(x) Λ [W(x)]' → [R(x)]')

How many ways can a committee of 2 be chosen from 4 men and 3 women and it must include at least 1 man?

C(7,2) = all committees possible C(3,2) = all committees with no men on it C(7,2) - C(3,2)

What are rational numbers? how are they represented?

Can be expressed as a fraction. Include integers and fractions or decimals. They are represented with a Q, exempel 1/2, 2/3, 4/7 eller 6.7

All drivers are happy only if they drive a Ford.

D(x): x is a driver H(x): x is happy F (x): x drives a Ford

Vad betyder detta? a∈A

Detta betyder att element a tillhör set A, bakåvänt e betyder "belongs to"

From a standard deck of 52 cards, in how many ways can 7 cards be drawn?

Detta är en fråga om combinationer 52C7 = 133,784,560

How many ways can we select a committee of three from 10?

Detta är en fråga om combinationer n = 10 r = 3 10! / (10 - 3)! * 3! = 120

How many distinct permutations can be made from thecharacters in the word MISSISSIPPI?

Detta är en fråga om permutationer M: 1 I: 4 S: 4 P: 2 11! / 4! * 4! * 2! = 34650

How many distinct permutations can be made from thecharacters in the word FLORIDA?

Detta är en fråga om permutationer n = 7! r = 7! 7! / 7! - 7! = 7!

Ten athletes compete in an Olympic event. Gold, silver andbronze medals are awarded to the first three in the event,respectively. How many ways can the awards be presented?

Detta är permutationer 3 objects pool of 10 P(10,3) 10! / (10! - 3!) = 10 * 9 * 8 svar: 720

How many ways can six people be seated on six chairs?

Detta är permutationer 6 objects pool of 6 P(6,6) 6! / 6! - 6! = 6! svar: 720

Hur många pokerhänder innehåller kort i samma färg? Tips: kortlek har 52 kort En hand i poker är 5 kort

En färg har 13 kort En poker hand innehåller 5 kort Var och en av de fyra färgerna har 13C5 = 1287 möjliga femkortshänder som alla är av samma färg. 4 * 1287 = 5148

Vad är en reflexive relation?

En reflexive = "arrows to self" Alltså for all elements x, (x måste han en relation till x) om du har settet (a,b) så måste a ha en relation till a, b ha en relation till b

How many ways can a committee of two women and three men be selected from a group of five different women and six different men?

For selecting two out of five women, we have C(5,2) ways = 10. For selecting three out of six men, we have C(6,3) ways = 20 Total number of ways for selecting the committee = 10*20 = 200

What is the smallest positive linear combination of x and y

It is the same as the gcd of x and y

What is natural numbers? how are they represented?

It is whole numbers. The numbers that we use when we are counting. positive integers They are represented with an N

Förklara skillnaden mellan permutationer och combinationer

Med permutationer så spelar varje liten detalj in, Alice, Bob och Charlie är annorlunda än Charlie, Bob och Alice Med combinationer så spelar detaljerna ingen roll, Alice, Bob och Charlie är samma som Charlie, Bob och Alice

Some people are rich and famous but not sick.

P (x): x is a person, R(x): x is rich, F (x): x famous, S(x): x is sick

None of the players scored a goal but all of them had a shot.

P (x): x is a player, S(x): x scored a goal, H(x): x had a shot

S = {1, 2, 4}, what is the cartesian product of set S with itself?

S * S = {(1,1), (1,2), (1,4), (2,1), (2,2), (2,4), (4,1), (4,2), (4,4)}

kolla på andra

S : 23, 12, 9, -3, 89, 54 n = 6

S(1) = 2 S(n) = 2S(n-1) for n >= 2 Vad blir S(1), S(2), S(3), S(4) och S(5)

S(1) = 2 S(2) = 4 S(3) = 8 S(4) = 16 S(5) = 32

Prove this by contradiction: There is no odd integer that can be expressed as the sum of three even integers

So we assume by contradiction that there is an odd integer that equals 2k +1, so a = 2k +1, where k belongs to a set of integers there also exast a = three even integers a = 2L + 2M + 2N, and l,m and n is even integers then a = 2(l + m + n) We assumed that a = 2k + 1 //odd but it shows that a is equal to a = 2(l + m + n) //even a cant be even and od the same time. therefore our assumption a = 2k + 1 is false, so then the original statement must be true

prove this by contradiction: If 3n + 7 is even, then n is odd

So we have P --> q the negation of this p-->q is p^q' so we assume p ^ q' (p and not q is true) in text this will be: if 3n + 7 is even, then n is even since n is even --> n = 2k then 3n + 7 = 3(2k) + 7 so 6k + 7 we want to get to this stage: so 6k + 7 = 2(3k + 3) + 1 so this shows that 3n + 7 is odd and we assumed that 3n + 7 is even Then we have our contradiction

Använd permutations och multiplikationprincipen för denna How many permutations of the letters ABCDEF contain the letters DEF together in any order?

The multiplication principle states that we can simply multiply the number of options in each category (screen size, memory, color) to get the total number of possibilities If DEF is considered as one letter, then we have 4 letters A B C DEF which can be permuted in 4! ways, DEF can be ordered by its letters in 3! ways.Hence, by the multiplication principle, total number of orderingspossible = 4!*3! = 24*6 = 144.

Is 1021 a prime number?

The value of (1021)^1/2 is justless than 32. So the primes we need to test are 2, 3, 5,7, 11, 13, 17, 19, 23, 29, 31. None divides 1021, so 1021 is prime.

What can we conclude if you don't win races but do train hard?

Then you don't eat good food

A child is allowed to choose one jellybean out of two jellybeans(one red and one black), and one gummy bear out of threegummy bears (yellow, green, and white). How many differentsets of candy can the child have?

There are 2x3=6 or 3x2=6 possible outcomes as seenfrom the following figures

Statement "b divides a" implies that ∃n ∈ Z such that a = nb, this is written as b|a. Let expression S be defined as "if b|a and a > 0, then b <= a a) Define your statement letters A, B, C b) write S with logical connectives and your statement letters

a) svar: A: b|a B: a > 0 C: b ≤ a b) svar: A ∧ B → C

prove by contradiction if a^2 is even, then a is even

by contradiction So we have P --> q the negation of this p-->q is p^q' so we assume p ^ q' (p and not q is true) in text this is: if a^2 is even and a is odd so a = 2k +1 then a^2 = (2k + 1)^2 (2k + 1)(2k + 1) 4k^2 + 2k + 2k + 1 4k^2 + 4k + 1 now we need to find 2( ) + 1 so 2(2k^2 + 2k) + 1 so now we have showed that a^2 is odd and our assumption is that it is even. so we now have proof by contradiction

For set S = {1,2,3} how many power sets can be made?

for a set with n elements, the power set has 2^n elements

Let A = {1, 2, 3, 5, 10} B = {2, 4, 7, 8, 9} C = {5, 8, 10} be subsets of S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Find : (A ∪ B) ∩ C'

{1,2,3,4,7,9}

{x | x is an integer and 4 < x < 9} Vad kommer x innehålla?

{5, 6, 7, 8}

{x | x is a month with exactly thirty days} Vad kommer x innehålla?

{April, June, September, November}

What is the predicate for each of the following sets? {2, 3, 5, 7, 11, 13, 17, ....}

{x | x is a prime number}