Which of the following is a multiplicative identity for a set of natural numbers?a.1b.0c.-1d.1/a

1. Break Down the Problem

To determine which option serves as the multiplicative identity for a set of natural numbers, we need to understand the concept of multiplicative identity.

2. Relevant Concepts

The multiplicative identity is a number that, when multiplied by any element of a set, leaves that element unchanged. In mathematical terms, for any number $a$ in a set, the multiplicative identity $e$ must satisfy: $a \cdot e = a$

3. Analysis and Detail

• Option a: 1
  Multiplying any natural number by $1$: $a \cdot 1 = a$ This holds true for all natural numbers.

• Option b: 0
  Multiplying any natural number by $0$: $a \cdot 0 = 0$ This does not satisfy the identity property, as it changes the value.

• Option c: -1
  The natural numbers do not include negative numbers, so this option is irrelevant in this context.

• Option d: $\frac{1}{a}$
  Although this would return 1 when $a$ is equal to 1, it does not generally apply to all natural numbers $a$. For instance, $2 \cdot \frac{1}{2} = 1$, which does not equal $2$.

4. Verify and Summarize

The only option that maintains the identity property for all natural numbers is 1. Therefore, after analysis of all options and verifying their validity against the definition of the multiplicative identity, we conclude:

Final Answer

The multiplicative identity for a set of natural numbers is a. 1.