Left Cosets are Equal iff Element in Other Left Coset/Proof 1

Theorem

Let $x H$ denote the left coset of $H$ by $x$.

Then:

$x H = y H \iff x \in y H$

Proof

\(\ds x H\)

\(=\)

\(\ds y H\)

\(\ds \leadstoandfrom \ \ \)

\(\ds x^{-1} y\)

\(\in\)

\(\ds H\)

Left Cosets are Equal iff Product with Inverse in Subgroup

\(\ds \leadstoandfrom \ \ \)

\(\ds x\)

\(\in\)

\(\ds y H\)

Element in Left Coset iff Product with Inverse in Subgroup

$\blacksquare$