   Chapter 3.1, Problem 3E

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# In Exercises 1 − 14 , decide whether each of the given sets is a group with respect to the indicated operation. If it is not a group, state a condition in Definition 3.1 that fails to hold.The set ℚ of all rational numbers with operation addition.

To determine

Whether the set of all rational numbers is a group with operation addition.

Explanation

Given information:

The set of all rational numbers with operation addition.

Explanation:

Let us check that the set of all rational numbers with operation addition is a group or not by using the definition given below:

Suppose the binary operation is defined for element of a set G. The set G is a group with respect to , provided the following conditions hold:

1. G is closed under . That is xG and yG imply that xy is in G.

2. is associative. For all x,y,z in G, x(yz)=(xy)z.

3. G has an identity element e. There is an e in G such that xe=ex=x for all xG.

4. G contains inverses. For each aG, there exists bG such that ab=ba=e.

First condition:

Let xy and ab, then xy+ab=xb+ayyb.

As the addition of rational numbers is a rational number, is closed under addition.

Second condition:

Let xy, ab and pq then

xy+(ab+pq)=xy+(aq+pbbq)=(xbq+y

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