Which of the following statements is/are true?

Question

Which of the following is/are True?

$S_1$: Set of Integers on addition operation is Monoid but not group.

$S_2$: Set of Integers on subtraction operation is Monoid but not group.

$S_3$: Set of Integers on multiplication operation is group but not Abelian group.

 

please elaborate on s2 statement.

Answer 1

S1: It is False

addition of Integer is closed

Addition of integer is associative ((1+2)+3)=(1+(2+3))

Identity property also satisfied 0+1=1, here identity element is 0

Inverse of 1 is (-1), 1+(-1)=0, So, inverse also satisfied

So, Integer addition will be a group

S2: It is also false

Associativity also not satisfied

((7-2)-3) not equal with (7-(2-3))

i.e.(5-3) not equal with (7+1)

but in monoid we have to satisfy associativity and identity property

S3: Also False

As, Integer multiplication doesnot satisfy inverse property

inverse of 2 will be 1/2

which is not an integer

So, Integer multiplication cannot form a group

Comments

Substactition is not closed always

thats not true

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yes, now see