Question

Q.1: Do the odd integer form a group w.r.t addition?
Q.2: Do the non-zero positive real number form a group w.r.t multiplication?

Answer 1

Answer on Question #58979 – Math – Abstract Algebra

Question

Do the odd integers form a group w.r.t addition?

Solution

Since odd integer $a, b$ can be written as follows:

$$a = 2n + 1, \quad b = 2m + 1, \quad n, m = 0, \pm 1, \pm 2, \dots$$

we obtain

$$a + b = 2(n + m + 1) \neq 2k + 1, \quad k = 0, \pm 1, \pm 2, \dots$$

Therefore, addition of two odd integers yields even integer. Closure axiom is not satisfied, therefore the odd integers do not form a group w.r.t addition.

Answer: the odd integers do not form a group w.r.t addition.

Question

Do the non-zero positive real numbers form a group w.r.t multiplication?

Solution

Given any non-zero positive real numbers: $a \in R^{+}, b \in R^{+}$.

We have to check group axioms w.r.t multiplication:

1. Closure:

We have $a \cdot b = c$, where $c \in R^{+}$ is some non-zero positive real number.

Axiom is satisfied.

2. Identity element:

We have $a \cdot 1 = 1 \cdot a = a$. Thus, identity element is $e = 1$. Axiom is satisfied.

3. Inverse element:

We have $a \cdot \frac{1}{a} = \frac{1}{a} \cdot a = e$. Thus inverse element is $a^{-1} = \frac{1}{a}$. Axiom is satisfied.

4. Associativity

As $a \cdot (b \cdot c) = (a \cdot b) \cdot c$ is obviously satisfied for any $a \in R^{+}, b \in R^{+}, c \in R^{+}$.

Axiom is satisfied.

Therefore, non-zero positive real numbers satisfy all group axioms w.r.t multiplication.

The non-zero positive real numbers form a group w.r.t multiplication.

Answer: The non-zero positive real numbers form a group w.r.t multiplication.

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