Subset of Subset Product

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Theorem

Let $\left({S, \circ}\right)$ be a magma.

Let $\mathcal P \left({S}\right)$ be the power set of $S$.

Let $X, Y, Z \in \mathcal P \left({S}\right)$.


Then:

$X \subseteq Y \implies \left({X \circ Z}\right) \subseteq \left({Y \circ Z}\right)$
$X \subseteq Y \implies \left({Z \circ X}\right) \subseteq \left({Z \circ Y}\right)$

where $X \circ Z$ etc. denotes subset product.


Proof

Let $x \in X, z \in Z$.

Then:

$x \circ z \in X \circ Z$ and $z \circ x \in Z \circ X$

Now:

$Y \circ Z = \left\{{y \circ z: y \in Y, z \in Z}\right\}$
$Z \circ Y = \left\{{z \circ y: y \in Y, z \in Z}\right\}$

But by the definition of a subset:

$x \in X \implies x \in Y$

Thus:

$x \circ z \in Y \circ Z$ and $z \circ x \in Z \circ Y$

and the result follows.

$\blacksquare$


Sources