The monoid of natural numbers under addition has the identity element:a.-1b.0c.1d.∞

Understanding the Monoid of Natural Numbers

1. Definition of a Monoid: A monoid is an algebraic structure that consists of a set equipped with an associative binary operation and an identity element.

2. Natural Numbers Under Addition: The set of natural numbers (denoted as $\mathbb{N}$) under the operation of addition is a common example of a monoid. The natural numbers are typically defined as $\{0, 1, 2, 3, ...\}$.

3. Identity Element in Addition: The identity element $e$ in a monoid must satisfy the condition $e + a = a + e = a$ for all elements $a$ in the set.

Analysis of Options

1. a. -1: This is not a natural number, hence cannot be the identity element.
2. b. 0: For $0$ to be the identity, it must satisfy $0 + a = a + 0 = a$ for all $a$. This holds true, as $0$ is in $\mathbb{N}$.
3. c. 1: While $1 + a = a + 1 = a + 1$ holds, it does not satisfy the identity condition for all $a$; thus, it cannot be the identity element.
4. d. ∞: This is not a member of natural numbers and does not apply to addition in this context.

Final Answer

The identity element of the monoid of natural numbers under addition is: b. 0.