Explanation Formula used: The equation of a line is of the form y = m x + c , where m is the value of the slope. Perpendicular Theorem: If two lines are perpendicular then the product of their slopes will be negative reciprocal of each other. If m is the slope of line y = m x + c and n is the slope of line perpendicular to y = m x + c then m = − 1 n . Pythagorean Theorem: In a right-angled triangle the square of the hypotenuse is equal to the sum of squares of the other two sides. Proof: Suppose that y = m 1 x + b 1 and y = − m 2 x + b 2 be the equations of two lines. The figure of y = m 1 x + b 1 and y = − m 2 x + b 2 is as follows: From the figure the vertical line is at x = 1 and Δ A B C is a right-angle triangle. Now, apply Pythagorean Theorem on Δ A B C , ( m 1 − m 2 ) 2 = a 2 + b 2 m 1 2 + m 2 2 − 2 m 1 m 2 =

Finite Mathematics & Its Applications (12th Edition)

12th Edition

ISBN: 9780134507125

Author: Goldstein

Publisher: PEARSON

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Chapter 1.2, Problem 88E

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To prove: The perpendicular property by using point-slope formula and the Pythagorean Theorem assuming that both the lines pass through the origin.

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Chapter 1.2, Problem 87E

Chapter 1.2, Problem 89E

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Q1) Classify the following statements as a true or false statements a. Any ring with identity is a finitely generated right R module.- b. An ideal 22 is small ideal in Z c. A nontrivial direct summand of a module cannot be large or small submodule d. The sum of a finite family of small submodules of a module M is small in M A module M 0 is called directly indecomposable if and only if 0 and M are the only direct summands of M f. A monomorphism a: M-N is said to split if and only if Ker(a) is a direct- summand in M & Z₂ contains no minimal submodules h. Qz is a finitely generated module i. Every divisible Z-module is injective j. Every free module is a projective module Q4) Give an example and explain your claim in each case a) A module M which has two composition senes 7 b) A free subset of a modale c) A free module 24 d) A module contains a direct summand submodule 7, e) A short exact sequence of modules 74.

************* ********************************* Q.1) Classify the following statements as a true or false statements: a. If M is a module, then every proper submodule of M is contained in a maximal submodule of M. b. The sum of a finite family of small submodules of a module M is small in M. c. Zz is directly indecomposable. d. An epimorphism a: M→ N is called solit iff Ker(a) is a direct summand in M. e. The Z-module has two composition series. Z 6Z f. Zz does not have a composition series. g. Any finitely generated module is a free module. h. If O→A MW→ 0 is short exact sequence then f is epimorphism. i. If f is a homomorphism then f-1 is also a homomorphism. Maximal C≤A if and only if is simple. Sup Q.4) Give an example and explain your claim in each case: Monomorphism not split. b) A finite free module. c) Semisimple module. d) A small submodule A of a module N and a homomorphism op: MN, but (A) is not small in M.

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Finite Mathematics & Its Applications (12th Edition)

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To prove: The perpendicular property by using point-slope formula and the Pythagorean Theorem assuming that both the lines pass through the origin.

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To prove: The perpendicular property by using point-slope formula and the Pythagorean Theorem assuming that both the lines pass through the origin.

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