Answer to Question #11320 in Discrete Mathematics for nitin mishra

Question

Let E = {2, 4, 6, …}. Then prove that (E, +) is a semigroup, where + is usual addition.

Answer 1

By definition E consists of all positive even integers.

We should verify
the following two properties:

1) a+b belongs to E for all a,b from
E.

Let a,b belong to E, so a and b a positive even integers, then so is
their sum a+b, i.e. a+b belongs to E



2) Operation + is
associative, i.e. (a+b)+c = a+(b+c)

This follows from the associativity
of addition operation "+" for all numbers.


Thus E is a semigroup.