Suppose we have $(\mathbb{C},+)$ and $i\mathbb{R}$. Since $(\mathbb{C},+)$ is abelian , we know that $i\mathbb{R}$ is a normal subgroup. Consider quotient group $(\mathbb{C},+)/{i\mathbb{R}}$. Then the set is $\lbrace[a+bi] |a,b \in \mathbb{R} \rbrace=\lbrace(a+bi)+i\mathbb{R} |a,b \in \mathbb{R} \rbrace$. My question is how do we know the quotient set contain what kind of elements ? In this case, how do we know the set contains equvalence classes of complex numbers ?
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There are several general notations for elements of a quotient group, and several specialized notations for specific cases. There are many groups that have a reasonable claim to be called a quotient group of $\mathbb{C}$ by $\mathbb{i}\mathbb{R}$. The particular group you write is a rather standard way of selecting a specific quotient of $\mathbb{C}$ by $\mathbb{i}\mathbb{R}$. It also fits into a rather common set-theoretic pattern for constructing interesting structures:
In this case, the information used to specify an element of the quotient group is by writing an element of $\mathbb{C}$ that is to be its pre-image. And so the elements of (the standard choice of) $\mathbb{C} / i \mathbb{R}$ are the cosets of $i \mathbb{R}$ in $\mathbb{C}$. |
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The answer seems to be: "this is how the quotient group is defined." If you have doubts whether is definition is meaningful, please present them. |
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