Subset of Subset Product

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Theorem

Let $\struct {S, \circ}$ be a magma.

Let $\powerset S$ be the power set of $S$.

Let $X, Y, Z \in \powerset S$.


Then:

$X \subseteq Y \implies \paren {X \circ Z} \subseteq \paren {Y \circ Z}$
$X \subseteq Y \implies \paren {Z \circ X} \subseteq \paren {Z \circ Y}$

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:

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

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