Use A Proof By Contradiction To Prove That The Sum Of Two Odd Integers Is Even CM (2023)

Mathematics College

Answers

Answer 1

Answer:

Sum of two odd integers is always even.

Step-by-step explanation:

Let m and n be two odd integers.

Since m and n are odd they can be written in the form m =2r + 1 and n = 2s + 1, where r and s are integers.

Let us suppose that their sum is not even.

m + n = (2r+1) + (2s + 1)

= 2r + 2s + 2

= 2(r+s+1)

= 2z

Thus, the sum of m and n can be written in the form 2z where z is an integer. But this is a contradiction to the fact that their sum is even.

Hence, our assumption was wrong and the sum of two odd integers is always even.

Related Questions

Let A and B be sets where |A1 = 9 and \B1 = 4. (a) What is the largest possible-value for (An BI? (b) What is the largest possible value for AU BI?
(c) What is the value of AU BI+IAN B?

Answers

Answer: a) 4, b) 13 and c) 17

Step-by-step explanation:

Since we have given that

Number of elements in A = 9

Number of elements in B = 4

a) What is the largest possible value for A∩B.

Since B has only 4 elements.

So, As largest possible, it can be B⊆A.

So, Largest possible value for A∩B = 4

(b) What is the largest possible value for AUB?

If suppose A∩B = Ф

n(A∩B) = 0

So, it becomes,

So, the largest possible value for A∪B = 13

(c) What is the value of AU B+A∩B?

|A∪B|+|A∩B|=13+4=17

Hence, a) 4, b) 13 and c) 17

Let X and Y be sets where [X] = 15 and \Y= 10. Let f:X + Y be a function. (a) If f is surjective then what is f(x))? (b) Is it possible for f to be injective? Be sure to explain your answer.

Answers

Answer:

Let be a function.

a) If f is surjective then each element of Y has a preimage in X, this means that .

Since Y has less elements that X, then f can not be injective because each element of X must have a image, then by the Pigeonhole principle at least one element of Y has more that one preimage.

How many 10-bit strings are there which: (a) Have weight 4? |(b) Have weight 4 and start with the substring 101?
|(c) Have weight 5 and start with 101 or end with 11 (or both)?|

Answers

Answer:

(a) 210

(b) 21

(c) 86

Step-by-step explanation:

(a)

We need to find how many 10-bit strings there are with only 4 bits = 1.

As the order does not matter, this number is a combination of 10 bits taken 4 at a time (those that are equal to 1)

So, there are 210 10-bit strings of weight 4

(b)

As the 10-bit strings start with 101, we need a 7-bit tail with only 2 bits =1.

The order does not matter, so this a combination of 7 taken 2 at a time

And there are 21 10-bit strings of weight 4 starting with 101

(c)

Let's compute first the number of 10-bit strings starting with 101 and having weight 5.

In this case, we need a 7-bit tail with only 3 bits =1.

Now, the 10-bit strings ending with 11. In this case, we need a 8-bit string with only 3 bits =1.

The number of 10-bit strings of weight 5 starting with 101 or ending with 11, would be 35+56 subtracting the strings starting with 101 and ending with 11, which were counted twice.

But these are 5-bit strings with only 1 bit =1, and there are 5.

So, the number of 10-bit strings of weight 5 starting with 101 or ending with 11 or both is

35+56-5 = 86.

Consider a bag containing five red marbles, three green ones, one transparent one, three yellow ones, and three orange ones How many possible sets of five marbles are there in which all of them red or green? sets Need Help? Read Tate Tutor

Answers

Answer: 56

Step-by-step explanation:

Given : Number of red marbles = 5

Number of green marbles = 3

Number of yellow marbles = 3

Number of orange marbles = 3

Number of red and green marbles = 5+3=8

Now the possible number of sets (combinations) of five marbles are there in which all of them red or green will be :-

Hence, the number of sets of five marbles in which all of them red or green=56

Answers

Answer:

Let denote the greatest common divisor of . We can prove this result as follows:

Step-by-step explanation:

The Bezout's identity establishes that if and only if for some integers .Since then we have that for some . Then,

Using the result of the Bezout's identity again we can concluide that .

Combinatorics: what is the coefficient of (a^2)(b^3)(c) in (2a - b + 3c)^6?

Answers

Answer:

The coefficient of a²b³c is -720

Step-by-step explanation:

Given:

Let 2a-b = x and 3c = y

General term of binomial expansion.

where, n=6 , r=1 ( because exponent of c is 1)

----------(1)

Now, we simplify (2a-b)⁵

The exponent of b is 3 and a is 2 .

If we take r=3 will get exponent of b is 3 and a is 2

So, put r=3

Substitute into equation (1)

Hence, The coefficient of a²b³c is -720

Consider a bag containing five red marbles, two green ones, one transparent one, four yellow ones, and two orange ones How many possible sets of five marbles are there in which none of them are red or green? sets Need Help? Tente Tutor

Answers

Answer: 21

Step-by-step explanation:

Given : A contains five red marbles, two green ones, one transparent one, four yellow ones, and two orange ones.

Total marbles =

Number of marbles except red or green =14-5-2=7

∵ Combinations of n things taken m at a time is given by :-

Now, the number of possible sets of five marbles are there in which none of them are red or green :_

Hence, the number of possible sets of five marbles are there in which none of them are red or green=21

In an arithmetic​ sequence, the nth term an is given by the formula an=a1+(n−1)d​, where a1 is the first term and d is the common difference.​ Similarly, in a geometric​ sequence, the nth term is given by 1an=a1•rn−1​, where r is the common ratio. Use these formulas to determine the indicated term in the given sequence. The 10th term of 40,10, 5/2, 5/8, ....

Answers

Answer:

Step-by-step explanation:

The first step to solving this problem is verifying if this sequence is an arithmetic sequence or a geometric sequence.

This sequence is arithmetic if:

We have that:

This is not an arithmetic sequence.

This sequence is geometric if:

This is a geometric sequence, in which:

The first term is 40, so

The common ratio is , so .

We have that:

The 10th term is . So:

Simplifying by 4, we have:

If a is an integer, prove that (14a +3,21a + 4) 1

Answers

Answer:

(14a + 3, 21a + 4) = 1

Step-by-step explanation:

Step-by-step explanation:

To prove that the greatest common divisor of two numbers is 1, we use the Euclidean algorithm.

1. In this case, and applying the algorithm we would have:

(14a + 3, 21a + 4) = (14a + 3, 7a + 1) = (1, 7a + 1) = 1

2. Other way of proving this statement would be that we will need to find two integers x and y such that 1 = (14a + 3) x + (21a + 4) y

Let's make x = 3 and y = -2

Therefore, (14a + 3, 21a + 4) = 1

In an arithmetic​ sequence, the nth term an is given by the formula an=a1+(n−1)d​, where a1 is the first term and d is the common difference.​ Similarly, in a geometric​ sequence, the nth term is given by 1an=a1•rn−1​, where r is the common ratio. Use these formulas to determine

the indicated term in the given sequence.

The 105th term of 1/2, 1, 3/2, 2,..

Answers

Answer:

105th term of given series is

Step-by-step explanation:

Given series is

As we can see,

Also,

hence, we can say given series is in arithmetic progression,

with common difference,

As given in question the nth term in A.P is given by

since we have to find the 105th term, so we can write

Hence, the 105th term of given series of A.P is .

A farmer looks out into the barnyard and sees the pigs and the chickens. He says to his daughter, "I count 153 heads and 346 feet. How many pigs and how many chickens are out there?"

Answers

Answer: There are 20 pigs and 133 chickens.

Step-by-step explanation:

We know that a pig has 4 legs and a chicken has 2 legs.

Let x be the number of pigs and y be the number of chickens , then we have the following system of linear equations:-

Multiply 2 on both sides of equation (1), we get

Eliminate equation (3) from equation (2), we get

Put x= 20 in (1), we get

Hence, there are 20 pigs and 133 chickens.

For a certain event, 817 tickets were sold, for a total of $1919. If students paid $2 per ticket and nonstudents paid $3 per ticket, how many student tickets were sold?

Answers

Answer:

532 student tickets were sold

Step-by-step explanation:

Let x be the no. of student's tickets sold

Let y be the no. of non student's tickets sold

We are given that 817 tickets were sold in total

So, --1

Cost of 1 student's ticket = $2

So, Cost of x student's ticket = 2x

Cost of 1 non - student's ticket = $3

Cost of y non - student's ticket = 3y

We are given that 817 tickets were sold, for a total of $1919

So, ---2

Solving 1 and 2

Substitute the value of y from 1 in 2

Substitute the value of x in 1

Hence 532 student tickets were sold

Why is the number 19/100 a rational number? A. It is the quotient of 100 divided by 19.
B. It is the quotient of 19 divided by 100.
C. It is the quotient of 10 divided by 9.
D. It is the quotient of 9 divided by 10.

Answers

Answer:

B. It is the quotient of 19 divided by 100

Step-by-step explanation:

A rational number is the quotient of two integers. 19/100 is the quotient of the integers 19 and 100.

Let S = {1, 3, 5, 7}. Define the set J = {2j^2 − 11 | j ∈ S}. List the elements of J.

Answers

Answer:

J={-9, 7,39,87}

Step-by-step explanation:

Given the statement "If I have the disease, then I will test positive." Show all work. (Discrete Mathematics) a) Write the converse.

b) Write the inverse.

c) Write the contrapositive.

d) Write the statement as a disjunction.

e) Write the negation.

Answers

Answer:

a) if I test positive, then I will have the disease

b) if I don't have the disease, then I won't test positive

c) if I don't test positive, then I won't have the disease

d) either I don't have the disease or I will test positive

e) I have the disease and I won't test positive.

Step-by-step explanation:

Having the statement: if m, then n.

a) the converse will be:

if n, then m.

m= I have the disease

n= I (will) test positive.

converse: if I test positive, then I (will) have the disease.

(It doesn't have to be true always).

b) the inverse will be:

if not m, then not n.

inverse: if I don't have the disease, I won't test positive.

c) the contrapositive will be:

if not n, then no m.

contrapositive: if I don't test positive, then I (will) not have the disease.

(It doesn't have to be true always).

d) disjunction:

We can rewrite if m, then n as: m⇒n, the disjunction will be:

m⇒n ≡ ¬m ∨ n (not m or n).

disjunction: either I don't have the disease or I will test positive.

e) negation:

the negation of m⇒n is ¬(m⇒n) ≡ ¬(¬m ∨ n) ≡ m∧¬n.

negation: I have the disease and I won't test positive.

I have the disease but I won't test positive.

(this is the only statement completely false).

If you had 454g of shrimp/sq ft in a tank with an 11 sq meter bottom area, how many total kilograms of shrimp do you have?

Answers

Answer:

total weight of the shrimp = 53.7549 kg.

Step-by-step explanation:

Given,

Weight of shrimp = 454 g/sq meter

area of bottom of tank = 11 sq meter

By converting meter into ft,

1 sq meter = 10.7639 sq feet

=> 11 sq meter = 11 x 10.7639 sq ft

= 118.4029 sq feet

So, the total area of the bottom of the tank = 118.4029 sq feet

According to question,

Weight of shrimp in 1 square ft = 454 g

so, the weight of shrimp in 118.4029 sq ft = 118.4029 x 454 g

= 53754.9166 g

Since,

\ 53754.9166\ g\ =\ 53754.9166\ g\times 10^{-3}\ kg" alt="=>\ 53754.9166\ g\ =\ 53754.9166\ g\times 10^{-3}\ kg" align="absmiddle" class="latex-formula">

= 53.7549 kg

So, the total weight of the shrimp will be 53.7549 kg.

The USS Enterprise was 1,123 feet in length. A. What is the scale for a model that is 30 inches long?

B. What is the scale for a model that is 2 feet long?

Answers

Answer:

The total length is 1123 feet.

a) if the model is 30 inches long, the first thing we need to do is write inches in feets, so 1 feet is equivalent to 12 inches.

so 30 inches are 30/12 = 2.5 feets.

the scale is the ratio between the feets in the the two lengths.

1123/2.5 = 449.2

so the scale is 1:449.2

so each feet in the model is equivalent to 449.2 feets in the enterprise.

b) if the model is 2 feet long, then:

1123/2 = 561.5

the scale is 1:561.5

so each feet in the model is equivalent to 561.5 feets in the enterprise.

You perform the calculation: 35920 / 172 on your calculator and its output is 208.837209. What is the answer with the correct number of significant figures? 208.837209 208 ООООО 208.84 I DON'T KNOW YET

Answers

Answer:

209

Step-by-step explanation:

When you have a product or a quotient the significant figures of the result is given with the same significant figures of the number that has the least significant figures.

35920 has four significant figures and 172 has three the answer will have 3 significant figures.

Then your answer is 209, it is because we have rounded.

Answer the following true or false. Justify your answer. (a) If A is a subset of B, and x∈B, then x∈A.
(b) The set {(x,y) ∈ R2 | x > 0 and x < 0} is empty.
(c) If A and B are square matrices, then AB is also square.
(d) A and B are subsets of a set S, then A∩B and A∪B are also subsets of S.
(e) For a matrix A, we define A^2 = AA.

Answers

Answer:

a) False

b) True

c) True

d) True

e) True

Step-by-step explanation:

a) Consider the sets and . Observe that A is a subset of B and but

b) Any number different of 0 can be positive and negative simultaneusly. Then doesn't exist such that . Then the set 0\; \text{and}\; x < 0}" alt="\{(x,y) \in \mathbb{R}^2 | x > 0\; \text{and}\; x < 0}" align="absmiddle" class="latex-formula"> is empty.

c) If the multiplication AB is defined and A and B are square matrices with A of size nxn, then B is the size nxn and the matrix AB is the size nxn.

d) Let A and B subsets of a set S. Since each element of A and B are in S then each element of is in S. Also, if , the and then . This shows that .

e) By definition AA=A^2

(1 point) Newton's law of cooling states that the temperature of an object changes at a rate proportional to the difference between its temperature and that of its surroundings.Suppose that the temperature of a cup of coffee obeys Newton's law of cooling. Let k>0k>0 be the constant of proportionality. Assume the coffee has a temperature of 190 degrees Fahrenheit when freshly poured, and 33 minutes later has cooled to 180 degrees in a room at 68 degrees.(a) Write an initial value problem for the temperature T of the coffee, in Fahrenheit, at time t in minutes. Your answer will contain the uknown constant k :dTdt=Equation EditorT(0)=Equation Editor(b) Solve the initial value problem in part (a). Your answer will contain the unknown constant k .T(t)=Equation Editor(c) Determine the value of the constant kk=Equation Editorminutes.(d) Determine when the coffee reaches a temperature of 150 degrees.Equation Editorminutes.

Answers

Answer:

a.

b.

c.

d. minutos

Step-by-step explanation:

a. Newton's law of cooling states that the speed with which a body is cooled is proportional to the difference between its temperature and that of the medium in which it is found. Then, the initial value problem is given by:

b. The differential equation obtained is a differential equation of separable variables:

c. After 33 minutes of serving the coffee has cooled to 180°:

d.

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