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Answered: An operator Tona Unitary if and # onto…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.1: Finite Permutation Groups

Problem 10TFE: True or False Label each of the following statements as either true or false. 10. A transposition...

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Let V be the set of all positive real numbers. Determine whether V is a vector space with the operations shown below. x+y=xyAddition cx=xcScalar multiplication If it is, verify each vector space axiom; if it is not, state all vector space axioms that fail.
True or False Label each of the following statements as either true or false. 2. If * is a binary operation on a nonempty set , then is closed with respect to *.
True or False Label each of the following statements as either true or false. The set of all bijections from A to A is closed with respect to the binary operation of composition defined on the set of all mappings from A to A.
6. a. Give an example of mappings and , different from those in Example , where is one-to-one, is onto, and is not one-to-one. b. Give an example of mappings and , different from Example , where is one-to-one, is onto, and is not onto.
Exercises Work exercise 5 using U=a. Exercise5 Let U=a,b. Define addition and multiplication in P(U) by C+D=CD and CD=CD. Decide whether P(U) is a ring with respect to these operations. If it is not, state a condition in Definition 5.1a that fails to hold. Definition 5.1a: Suppose R is a set in which a relation of equality, denoted by =, and operations of addition and multiplication, denoted by + and , respectively, are defined. Then R is a ring with respect to these operation if the following conditions are satisfied : 1) R is closed under addition: xR,yRx+yR 2) Addition in R is associative: (x+y)+z=x+(y+z)x,y,zR 3) R contains an additive identity 0: x+0=0+x=xxR 4) R contains an additive inverse: for each x in R, there exists x in R such that x+(x)=(x)+x=0. 5) Addition in R is commutative: x+y=y+xx,yR 6) R is closed under multiplication: xR,yRxyR 7) Multiplication in R is associative: (xy)z=x(yz)x,y,zR 8) Two distributive laws holds in R: x(y+z)=xy+xz and (x+y)z=xz+yz x,y,zR
7. a. Give an example of mappings and , where is onto, is one-to-one, and is not one-to-one. b. Give an example of mappings and , different from example , where is onto, is one-to-one, and is not onto.
True or False Label each of the following statements as either true or false. 10. A transposition leaves all elements except two fixed.
13. Consider the set of all nonempty subsets of . Determine whether the given relation on is reflexive, symmetric or transitive. Justify your answers. a. if and only if is subset of . b. if and only if is a proper subset of . c. if and only if and have the same number of elements.

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An operator Tona Unitary if and # onto itself.' only 2 Hilbert space H u if is an sometric of