Question #11320

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

Expert's answer

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.

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.

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