Determine if the following sets are closed under the given operations. a) The set of odd intergers: addition b) The set of even intergers: addition c) The set of rational numbers: mulitplication...

Determine if the following sets are closed under the given operations.

a) The set of odd intergers: addition

b) The set of even intergers: addition

c) The set of rational numbers: mulitplication

Enter three answers (yes or no) separated by commas. 

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aruv's profile pic

aruv | High School Teacher | (Level 2) Valedictorian

Posted on

Let  S be a set. Let `a,b inS`  , if `a**b in S`  then S is closed with respect to operation `**` .

1.

`S={.........,-7,-5,-3,-1,1,3,5,7,.......}`

let  a and b are two arbitrary element of S. Then a+b is an even number because sum of two odd is even number. Therefore a+b does not belongs to S. So S is not closed under addition.

2.

`S={..........,-6,-4,-2,2,4,6,............}`

let  a and b are two arbitrary element of S. Then a+b is an even number because sum of two even is even number. Therefore a+b does  belongs to S. So S is  closed under addition.

3.

`S={p/q:(p,q)=1,q!=0,and p,qin R}`

let `a/b and c/d`  are two arbitrary elements of S.

`(a/b) xx (c/d)=(ac)/(bd)` , `b!=0,d!=0=>bd!=0`

ac and bd are real number.Thus `(ac)/(bd)` does belongs to S. So S is closed under multiplication.

llltkl's profile pic

llltkl | College Teacher | (Level 3) Valedictorian

Posted on

A set is said to be closed for an operation if the result of that operation is an element of the same set.

a) If you add two odd integers, say, 1 and 3 you get an even integer (4 in this case), which is not an element of that set. So the set of odd integers is not closed under addition.

b) If you add two even integers you always get an even integer, so the set of even integers is closed under addition.

c) If you multiply two rational numbers you  get a rational number as the product. Thus, multiplication of (3/4)*(9/10) yields 27/40 as the product, which is also a rational number. So the set of rational numbers is closed under multiplicaton.

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