ThmDex – An index of mathematical definitions, results, and conjectures.

Result R2062 on D70: Set difference

Antitonicity of subtracting from the same set

Formulation 0

Let $X, Y, Z$ each be a D11: Set such that

(i)	\begin{equation} Y \supseteq Z \end{equation}

Then \begin{equation} X \setminus Y \subseteq X \setminus Z \end{equation}

Formulation 1

Let $X, Y, Z$ each be a D11: Set.

Then \begin{equation} Z \subseteq Y \quad \implies \quad X \setminus Y \subseteq X \setminus Z \end{equation}

Proofs

Proof 0

Let $X, Y, Z$ each be a D11: Set such that

(i)	\begin{equation} Y \supseteq Z \end{equation}

If $X \setminus Y$ is empty, then the claim holds due to R7: Empty set is subset of every set.

Suppose thus that $X \setminus Y$ is not empty and let $x \in X \setminus Y$. Then $x \in X$ and $x \not\in Y$. Since $Z$ is a subset of $Y$, it must be true that $x$ is not in $Z$ either. But now, by definition, $x \in X \setminus Z$. Since $x \in X \setminus Y$ was arbitrary, the claim follows. $\square$