Question

Q. Do the non zero positive rational no. form a group w.r.t multiplication.

Answer 1

Answer on Question #58448 – Math – Abstract Algebra

Question

Do the non zero positive rational no. form a group w.r.t multiplication?

Solution

1) Closure: $a, b \in \mathbb{R} \setminus \{0\} \Rightarrow a^*b \in \mathbb{R} \setminus \{0\}$;

2) Associativity: $a, b, c \in \mathbb{R} \setminus \{0\} \Rightarrow (a \bullet b) \bullet c = a \bullet (b \bullet c)$;

3) Identity element: $1 \in \mathbb{R} \setminus \{0\}$; $a \bullet 1 = 1 \bullet a = a$;

4) Inverse element: for each $a \in \mathbb{R} \setminus \{0\}$ there exists $a^(-1) = 1/a$ such that $a \bullet b = b \bullet a = 1$.

Answer: Yes.

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