Can you answer letter D, no. 1? D. Determine whether or not each of the following definitions of * given below does give a binaryoperation on the given set. In the event that * is not a binary operation, state whether condition1 (well definedness, condition 2 (closure property), or both of these conditions are violated.1.On Z*, define * by a * b = a/b2. On R- {0}, define * by a *b = abE. For each binary operation * defined on a set below, say whether or not * gives a group structureon the set. If no group results, give the axiom that does not hold.1. (R* , * ) where * is defined by a * b = ab2. (Z, * ) where * is define by a * b = a - b.F. Let a e G where G is a group. What shall you show to prove that a= q?

Solution of D part (1): (1) The set ℤ+ is ℤ+=1,2,3,... The operation * is defined as a*b=ab

D. Determine whether or not each of the following definitions of * given below does give a binary operation on the given set. In the event that * is not a binary operation, state whether condition 1 (well definedness, condition 2 (closure property), or both of these conditions are violated. 1. On Z*, define * by a * b = a/b

D. Determine whether or not each of the following definitions of * given below does give a binary operation on the given set. In the event that * is not a binary operation, state whether condition 1 (well definedness, condition 2 (closure property), or both of these conditions are violated. 1. On Z, define by a * b = a/b

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.1: Definition Of A Group

Problem 43E: Write out the elements of P(A) for the set A={ a,b,c }, and construct an addition table for P(A)...

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9. The definition of an even integer was stated in Section 1.2. Prove or disprove that the set of all even integers is closed with respect to a. addition defined on . b. multiplication defined on .
12. (See Exercise 10 and 11.) If each is identified with in prove that . (This means that the order relation defined in Exercise 10 coincides in with the original order relation in . We say that the ordering in is an extension of the ordering in .) 11. (See Exercise 10.) According to Definition 5.29, is defined in by if and only if . Show that if and only if . 10. An ordered field is an ordered integral domain that is also a field. In the quotient field of an ordered integral domain define by . Prove that is a set of positive elements for and hence, that is an ordered field. Definition 5.29 Greater than Let be an ordered integral domain with as the set of positive elements. The relation greater than, denoted by is defined on elements and of by if and only if . The symbol is read “greater than.” Similarly, is read “less than.” We define if and only if. As direct consequences of the definition, we have if and only if and if and only if . The three properties of in definition 5.28 translate at once into the following properties of in . If and then . If and then . For each one and only one of the following statements is true: . The other basic properties of are stated in the next theorem. We prove the first two and leave the proofs of the others as exercises.
Write out the elements of P(A) for the set A={ a,b,c }, and construct an addition table for P(A) using addition as defined in Exercise 42. (Sec. 1.1,7c) Sec. 1.1,7c 42. For an arbitrary set A, the power set P(A) was defined in Section 1.1 by P(A)={ XXA }, and addition in P(A) was defined by X+Y=(XY)(XY) =(XY)(YX) Prove that P(A) is a group with respect to this operation of addition. If A has n distinct elements, state the order of P(A).
4. Let be the relation “congruence modulo 5” defined on as follows: is congruent to modulo if and only if is a multiple of , and we write . a. Prove that “congruence modulo ” is an equivalence relation. b. List five members of each of the equivalence classes and .