1)Let S be the set of all students at a college. Define a relation on the set S by the rule that two people are related if they live less than 2 miles apart. Is this relation an equivalence relation on S? Justify your answer.

2) Define another relation on the set S from problem 5 by defining two people as related if they have the same classification (freshman, sophomore, junior, senior or graduate student). Is this an equivalence relation on S? Justify.

1)Let S be the set of all students at a college. Define a relation on the set S by the rule that two people are related if they live less than 2 miles apart. Is this relation an equivalence relation on S? Justify your answer.

2) Define another relation on the set S from problem 5 by defining two people as related if they have the same classification (freshman, sophomore, junior, senior or graduate student). Is this an equivalence relation on S? Justify.

what do you understand by parttion values? write a short note on the role of parttion values in real life situations where these can be applied.

Find the zeroes and their order for the following. (Use Taylor series or derivatives)

sin(z)

A 2 liter tank of water contains 3 grams of salt at time t = 0 (in minutes). Brine with concentration 3t grams of salt per liter at time t is added at a rate of one liter per minute. The tank is mixed well and is drained at 1 liter per minute. At what positive time is there a minimum amount of salt and what is that amount?

Problem 11:

A summer resort took a poll of its 350 visitors to see which summer activities people preferred. The results are as follows:

112 of the resorts visitors liked to surf

121 of the resorts visitors liked to Jet Ski

145 of the resorts visitors liked to swim

18 of the resorts visitors liked to surf and Jet Ski

30 of the resorts visitors liked to surf and swim

25 of the resorts visitors liked to Jet Ski and swim

13 liked to participate in all three activities.

1. Draw a Venn Diagram:

2. How many people liked to do none of these activities?

3. How many people liked to surf and swim, but not Jet Ski?

4. How many people like to Jet Ski only?

5. How many people liked to surf or Jet Ski?

6. How many people did not like to swim?

f(x)= 9x^4-2x^3-36x^2+8x/3x^3+x^2-14

-Factor the numerator and denominator of f(x) completely. -Write the domain of f(x) in interval notation. -Locate all hole(s), if any, and write them in the form of coordinate pairs. -Locate all vertical asymptote(s), if any, and give their equations in the form x = c. For each one, describe what happens to f(x) as x approaches c from the left(-), and as x approaches c from the right (+). -Locate the horizontal/slant asymptote, if any, and give its equation in the form y = b (or y = mx+b). -Locate all x- and y-intercepts of f(x), if any, and give as coordinate pairs. -Construct a sign diagram and use test points to determine on which intervals f(x) is positive and negative. -Use all of this information to draw a sketch of the graph. Label all asymptotes, holes, and intercepts, as well as axes and tick marks.

Prove whether or not the set S is countable

a. S= {irrationals}

b. S= {terminating decimals}

c. S= [0, .001)

d. S= Q(rationals) x Q(rationals)

e. S= R(real numbers) x Z(integers)

Let f : N → N and g : N → N be the functions defined as ∀k ∈ N f(k) = 2k and g(k) = (k/2 if k is even, (k + 1) /2 if k is odd).

(1) Are the functions f and g injective? surjective? bijective? Justify your answers.

(2) Give the expressions of the functions g ◦ f and f ◦ g?

(3) Are the functions g ◦ f and f ◦ g injective? surjective? bijective? Justify your answers.

(Discrete Math) Prove that the equation 2x² + y² = 14 has no positive integer solutions.

REAL ANALYSIS I

Prove the following exercises (please show all your work)-

Exercise 1.1.2: Let S be an ordered set. Let A ⊂ S be a nonempty finite subset. Then A is bounded. Furthermore, inf A exists and is in A and sup A exists and is in A. Hint: Use induction.

Exercise 1.1.9: Let S be an ordered set and A is a nonempty subset such that sup A exists. Suppose there is a B ⊂ A such that whenever x ∈ A there is a y ∈ B such that x ≤ y. Show that sup B exists and sup B = sup A.

1. Find 4 examples from an outside source of a real-world scenario that involves a sequence.
2. Define the sequence in three different ways.
3. Determine if the sequence is arithmetic, geometric, or neither.
4. Discuss which representation of the sequence is best in this real-world scenario, and why.

A ​$257 comma 800 mortgage for 30 years for a new home is obtained at the rate of 7.8​% compounded monthly. Find​ (a) the monthly​ payment, (b) the interest in the first ​payment, (c) the principal repaid in the first​ payment, and​ (d) the finance charge. ​(a) The monthly payment on the mortgage is ​$ nothing. ​(Round the final answer to two decimal places as needed. Round all intermediate values to six decimal places as​ needed.) ​(b) The interest in the first payment is ​$ nothing. ​(Round the final answer to two decimal places as needed. Round all intermediate values to six decimal places as​ needed.) ​(c) The principal repaid in the first payment is ​$ nothing. ​(Round the final answer to two decimal places as needed. Round all intermediate values to six decimal places as​ needed.) ​(d) The finance charge is ​$ nothing.

A stock's price follows a lognormal model. You are given:

(i) The current price of the stock is 105.

(ii) The probability that the stock's price will be less than 98 at the end of 6 months is 0.3483.

(iii) The probability that the stock's price will be less than 115 at the end of 9 months is 0.7123.

Calculate the expected price of the stock at the end of one year.