Decide whether each of the given sets is a group with respectto the indicated operation. If it is
not a group, state a conditionthat failes to hold with this definition:
G is a group with respect to the binar operation *provided:
1) G is closed
2) * is associative
3) G has an identity element
4) G contains inverses
Are my answers correct, if not, canyou give a detailed explanation why. please thank you for
yourtime.
problem #1)
(a) The set of all rational numbers with operationaddition. I said TRUE
(b) The set of all irrational numbers with operationaddition. I said FALSE, noinverse
(c) The set of all positive rational numbers with operationmultiplication. I said TRUE
(d) The set E of all even integers with operation addition.I said TRUE
Decide whether each of the given sets is a group with respectto the indicated operation. If it is
not a group, state a conditionthat failes to hold with this definition:
G is a group with respect to the binar operation *provided:
1) G is closed
2) * is associative
3) G has an identity element
4) G contains inverses
Are my answers correct, if not, canyou give a detailed explanation why. please thank you for
yourtime.
problem #1)
(a) The set of all rational numbers with operationaddition. I said TRUE
(b) The set of all irrational numbers with operationaddition. I said FALSE, noinverse
(c) The set of all positive rational numbers with operationmultiplication. I said TRUE
(d) The set E of all even integers with operation addition.I said TRUE
Solution
a) good
b) irrationals with respect to addition is not a groupbecause it violates 1,3. It is not closed(-
2+2=0 and 0 is not irrational, and there is noirrational identity element.)
c) good
d) good
.
Nice work..
Decide whether each system is a group if not a group identify all p.pdf
1. Decide whether each system is a group? if not a group identify all properties that are not
satisfied.
4). {0}; addition
6).{0}; subtraction
8). {0,1}; multiplication
10). {-1,0,1}; addition
12).integers;multiplication
14). counting numbers; addition
16). even integers; addition
18). nonzero rational numbers; multiplication
20). explain why a group based on the operation of ordinary addition of numbers must contain
the element 0.
Solution
4). {0}; addition Yes. 0+0=0. 6).{0}; subtraction Yes. 0-0=0. 8). {0,1};
multiplication Yes. 0*0=0, 0*1=0, 1*0=0. 1*1=1. 10). {-1,0,1}; addition No. 1+1=2 And 2 is not
an element of the set. 12).integers;multiplication Yes. Any multiplication of two integers is again
an integer. The identity-element is 1 since n*1=n. 14). counting numbers; addition Yes. Any
sum of two counting numbers is again a counting number. The identity-element is 0. 16). even
integers; addition Yes. Any sum of two even integers is again an even integer. The identity-
element is 0. 18). nonzero rational numbers; multiplication Yes. Any multiplication of two
nonzero rational numbers is again a rational number: Let a=b/c and d= e/f then
a*d=(b/c)*(e/f)=(b*e/c*f) which is again a rational number. 1 is the identy-element. 20). explain
why a group based on the operation of ordinary addition of numbers must contain the element 0.
If it does not contain the element 0, there is no element b for which a+b=a.
