Question: Let A be a set with |A| = n, and consider the order for the listing of its elements as fixed. For R A

Let A be a set with |A| = n, and consider the order for the listing of its elements as fixed. For R ⊆ A × A, let M (R) denote the corresponding relation matrix.
(a) Prove that M (R) = 0 (the n × n matrix of all 0's) if and only if R = ∅.
(b) Use the result of Exercise 11, along with the Principle of Mathematical Induction, to prove that M(Rm) = [M(R)]m, for all m ∈ Z+.

Step by Step Solution

★★★★★

3.40 Rating (150 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

a If MR 0 then x y A x y R Hence R Conversely if MR 0 then x y A where xRy Hence ... 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 (7838).docx

120 KBs Word File

Students Have Also Explored These Related Linear Algebra Questions!

Let A be the set of students who live within one mile of school and let B be the set of students who walk to classes. Describe the students in each of these sets. a) A B b) A B c) A - B d) B A

Let A be the set of students at your school and B the set of books in the school library. Let R1 and R2 be the relations consisting of all ordered pairs (a, b), where student a is required to read...

Let n Z+ with n > 1, and let A be the set of positive integer divisors of n. Define the relation R on A by x R y if x (exactly) divides y. Determine how many ordered pairs are in the relation R when...

We have n colored pebbles, where each pebble is either red or green or blue. We want to order the pebbles in a straight line of length n such that no two adjacent pebbles will have the same color....

Product notation. In case this hasn't come up in your classes yet, given {a,..., an} CR we use the following notation: n IIaaa an as a shorthand to mean the multiplication of all the elements {a,...,...

Question 2: Let n 2 be an integer. Determine the number of strings consisting of n characters, where each character is an element of the set {a, b, 0}. Let S be a set consisting of n elements....

A. Let U = (-00, co) be the universal set and M = [-3, 7) and W = [-5,4). In interval notation, determine the following. 1. MOW 4. W - M 2. Mow 5. M 3. M - W 6. W B. Let U = {x x is a lowercase...

In a Hopfield neural network configured as an associative memory, with all of its weights trained and fixed, what three possible behaviours may occur over time in configuration space as the net...

Make sense of the thoughts of static and dynamic chains, giving an illustration of a circumstance in which they contrast (for example reference to some unacceptable one would get to the wrong...

Given the following data from a recent Comparative Competitive Efforts page in the CIR: Industry Your Company Average vs. Ind. Avg. $250 -6.0% 5.1 -2.0% -5.3% -3.2% -25.7% -25.0% -25.8% -15.3% -48.2%...

Sarah will receive $10,000 in 3 years and $15,000 in 5 years. The compound discount rate is 6.5% per year. Calculate the present value of today of the payments. At an annual effective interest rate...

4. (a) Name the process by which energy is made on the Sun and explain the source of the energy.

The chapter states, Firms will supply all those goods that provide consumers with a marginal benefit at least as great as the marginal cost of producing them. A student objects to this statement,...

A gazebo in the shape of a regular octagon has equal sides of 9 feet and an apothem of 10.9 feet. a. If one side of a gazebo is open, and the other sides have a railing, find the cost of the railing...

A rectangular room measures x feet by y feet, where x

Giuliana is having one wall of her den covered in brick. She needs to find the area of the wall, excluding the doors and the windows. A diagram of the room is shown. a. Giuliana creates a scale...

computech Corporation is expanding rapidly and currently needs to retain all of its earnings hands it does not pay dividends however investors expect computech to begin paying dividends beginning...

The central economic characteristic of a city's base activities is that they: give the city influence in the larger world are industrial bring in income from the larger world bring recognition and...

In hypothesis testing we set alpha, the significance level, which limits the likelihood of making a Type 1 error if the null hypothesis were true. When does a type 2 error occur? Give a real world...