Question: For any set A , let P(A) denote the set of all partitions of A, and let E(A) denote the set of all equivalence

For any set A ≠ ∅, let P(A) denote the set of all partitions of A, and let E(A) denote the set of all equivalence relations on A. Define the function f: E(A) → P(A) as follows: If R is an equivalence relation on A, then f(R) is the partition of A induced by R. Prove that f is one-to-one and onto, thus establishing Theorem 7.8.

Step by Step Solution

★★★★★

3.41 Rating (173 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

Let P U i I A i be a partition of A Then E iI A i A i is an equivalence relati... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Document Format (1 attachment)

954-M-L-A-L-S (7886).docx

120 KBs Word File

Students Have Also Explored These Related Linear Algebra Questions!

Let U denote the set of all points in and on the unit square shown in Fig. 7.29. That is, U = {(x, y) | 0 x 1, 0 y 1}. Define the relation R on U by (a, b) R (c, d) if (1) (a, b) = (c, d), or (2)...

Show that any equivalence relation on a set X partitions X.

Let X be any set. Let F(X) denote the set of all functions on X. Show that F(X) is a linear space.

Let A be a nonempty set and let P be a partition of A. Define a relation R (corresponding to P) on A by (a) Let A = {1,2,3}. Write down ALL possible partitions of A. For each of the partition P, use...

Explain the following 1. [Externalities and car accidents] Consider an economy with two individuals i = [1, 2) with the following quasi-linear utility function uj(s', q') = v'(s') + ow where s'...

Question 2. Suppose we have a coin (perhaps an unfair one) and we want to estimate the probability of flipping the coin and having it land heads up. Let p = P(H) denote the (unknown) population...

Please answer all the question with a clear explanation, thank you for your work! Consider the algorithm for building a CART model in the case of regression. Following and ex- panding on the notation...

So I currently have this problem that I need to solve, but I am very stuck on the actual solution. It revolves around the algorithm for building a CART model in case of regression. I've added the...

Hello I don't think the answers I got were correct for the paper. For instance, the second question obviously have an answer of A by the distributive and implication law. But the tutor gave an answer...

s1 educated (SSE) student for every three public school educated (PSE) students. Reasoning that students are not very dissimilar from threads, he suggests the following entry and exit routines be...

Big owns 80% of Little. At acquisition in 2021, all of Little's assets and liabilities had fair values equal to book value, and there was no goodwill (i.e., there was no differential). At the...

Potassium dichromate (K 2 Cr 2 O 7 ) is to be recovered from a 21 wt% aqueous solution in a continuous crystallization operation. The solution is joined by a recycle stream and fed to a vacuum...

5, 7, 10 B. J. and Carolyn Grace are full-time employees. B. J. is an elementary school teacher, and Carolyn is a registered nurse at a hospital. During the year, they incur the following...

During the eighteenth century, the American colonies had much more land per farmer than Europe. As a result, the price of labour in the colonies was much higher relative to the price of land than it...

A cable TV/Internet/phone provider charges Janet $90 per month for all three services. In addition, Janet's monthly bill for cell phone calls and text messages averages $77 per month. a. What does...

Text-Time charges $25 for a texting plan with 300 text messages included. If the customer goes over the 300 messages, the cost is $0.10 per message. They have an unlimited plan for $48 per month. a....

Show that the risk-neutral probability of a call finishing in the money is phi (d_2, (T, S_0, K, sigma, r)). Show that the risk-neutral probability of a call finishing in the money is phi (d_2, (T,...

find the mean, median and mode for one of the variables.Which of these measures of central tendency do you think is the best choice for "average" and why?Find the range and standard deviation...

how to solve the logistic model using R programming what are the code to use to produce the 1 st and 2 nd using R programming please provide the code. data is following: