### Equivalence Relations

H is the relation on the set of all people given by H = {(a,b)|a and b are the same height}. Is H an equivalence relation? Explain your answer.

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H is the relation on the set of all people given by H = {(a,b)|a and b are the same height}. Is H an equivalence relation? Explain your answer.

Show that ((p ══ > ~ q ) ^q) ══ > ~ p, This read (p implies not q AND q) implies not p

Given the following data set: 8, 7, 6, 9, 12 Find its standard deviation______________________

1. Identify the rule of inference used in the following: If it rains today, the flood gates will open. The flood gates did not open today. Therefore, it did not rain. a. modus tollens b. hypothetical syllogism c. modus ponens d. disjunctive syllogism 2. Identify the rule of inference used in the following: If I work all

Sort the given sequence of numbers using merge sort. Draw the splitting and merging trees for each application of the procedure. -1, 0, 2, -2, 3, 6, -3, 5, 1, 4

List the vertices of tree in attachment lib1312062.jpg, when they are visited in a preorder traversal and in a postorder traversal.

1. By using the Pigeonhole Principle, we can show that if you take six classes in a term and classes do not meet on the weekend, then at least three of the classes must meet on the same day. True False 2. By using the Pigeonhole Principle, it can be shown that if you are paid bi-weekly (every two weeks) duri

1) If f is the mod-5 function, compute each of the following. a) f(17) b) f(48) c) f(169) 3) Convert (1011101)2 to base 16 (i.e., hex) 4) Find the sum of products expansion of this Boolean function F(x,y) that equals 1 if and only if x = 1. Note: one can write out the phrase "y complement" to represent the notation f

(See attached file for full problem description). --- Give a recursive definition of a) the sequence {an}, n=1,2,3,...if i. an = 1+(-1)n ii. an = n2 b) of the set of ordered pairs of positive integers S = {(a,b) | a є Z+, b є Z+, and 3 |(a+b)}.

Perform operation and write the result in standard form (5 - 4i) / (3 - 2i)

(See attached file for full problem description) --- Consider the RSA encryption system given by p=43,q=59, and e=13 i) Find d such that ed ≡ 1 (mod (p-1)(q-1)) ii) Decode the message : 1552 2069 1178 1637 1975 Using the convention A = 00, B = 01, ..., Z = 25 ---

Find all common solutions to the congruences (in the following notations the = is meant to be a congruence symbol) x=2(mod 3), x=1(mod 4), x=3(mod 5), x=4(mod 7).

I) Show that x^3 is O(x^4) but x^4 is not O(x^3) ii)Show that xlnx is O(x^2) but x^2 is not O(xlnx) iii)Show that a^x O(b^x) but b^x is not O(a^x) if 0 < a < b (0 = zero) iv)Show that 1^k + 2^k+...+n^k is O(n^(k+1)) for every positive integer k

(See attached file for full problem description with proper symbols) --- 2. Let f(x) = x2 +1 and g(x) = {x+1, x> =3; x-1, x<3 so both f and g map R into Find the formula for a. (f+g)(x) b. (f .g)(x) c. (f o g)(x) d. (g o f)(x) 3. Let A = {a,b,c,d} and B = {1,2,3} and let f : A  B be a function . Let g : Z

We worked on the attached problems today in class I am now trying to work through them again for understanding and I am not getting very far. My skills in discrete mathematics are not such that I can work through these on my own effectively. 3. Seven points are located in a plane. List the possible numbers of lines determi

If we observe a random sample X1...Xn from a distribution with the pdf... Obtain the MLE of mu. Is it unbiased? Please see attached.

The plot below is the observed values of two random variables X and Y. What is the sign and a rough magnitude of the Corr (X,Y)? Please see attached.

1. How do I prove that in a group of n people there are two people with the same number of acquaintances within the group? 2. Prove that given a sequence of twelve integers, a1, a2, ...,a12, there is a subsequence aj, aj+1, ..., ak where 12 divides ∑kn= aa n. 3. A scrape of paper is found in an old desk that read:

30. a. Compute the covariance for X and Y in Exercise 22 (see attachment) b. Compute ρ (correlation coefficient) for X and Y in the same exercise. Please see the attached file for the fully formatted problems.

When a pair of dice is rolled, the total will range from 2 (1,1) to 12 (6,6). It is a fact that some numbers will occur more frequently than others as the dice are rolled over and over. Why will some numbers come up more frequently than others? Each die has six sides numbered from 1 to 6. How many possible ways can a numb

48. Suppose that only 25% of all drivers come to a complete stop at an intersection having flashing red lights in all directions when no other cars are visible... (see attachment for complete question)

On the first day of math class, 20 people are present in the room. To become acquainted with one another, each person shakes hands just once with everyone else. How many handshakes take place?

Let p represent some false statement and let q represent some true statement. What is the truth value of each of the following: a. ~(~p v q) b. ~(p v q) c. (~p) v q

1. True or False. In a multinomial experiment, all outcomes of each trial can have several categories. 2. In finding Expected Frequency you multiply what divided by what?

Let S be the set of all strings of a's and b's. Let R be the relation on S defined by....x and y begin with different symbols. Determine and prove your answers, whether or not R is reflexive, symmetric or transitive.

Construct addition and multiplication tables for arithmetic modulo 11. For example, 7 + 8 mod 11 is 4 and 7*8 mod 11 is 1 so the entry in row 7, column 8 would be 4 for the addition table and 1 for the multiplication table. Use your tables to solve each of the following congruences: a. 3x+2≡8 (mod11) b. 3x-5≡2 (m

Let S ne the set of all strings of a's and b's. Let R be the relation on S defined by... x and y begin with different symbols. Determine, and prove your answers, whether or not R is reflexive, symmetric or transitive. Please see the attached file for the fully formatted problems.

Prove or disprove (find a counterexample) : If A C B and f : A --> B is an onto function (the range of f is all of B), then f is one-to-one and A =B. Please see the attached file for the fully formatted problem.

Let f(x) = x^2 + x - 12. Consider f as a function from the reals to the reals. Is f one-to-one? Is f onto? What is the range of f? See the attached file.

The joint density of X and Y is given by... Are X and Y independent? What if f(x,y) were given by...? Please see attached.