For $Y \subseteq X$, is it true that $X^{\complement}\cap Z^{\complement} \subseteq Y^{\complement} \cap Z^{\complement}$?

Question

Given $$X^{\complement} \cap Z^{\complement} \subseteq Y^{\complement} \cap Z^{\complement},$$ applying De Morgan's laws, it changes to $$ (X \cup Z)^{\complement} \subseteq (Y \cup Z)^{\complement}.$$ If I am not mistaken, $A^{\complement} \subseteq B^{\complement} \implies B \subseteq A$. Using this, $$ Y \cup Z \subseteq X \cup Z.$$ Since it has been given that $Y \subseteq X$, the above statement should naturally follow.

This question was taken from the 2018 entrance test for PGDBA. The above reasoning was what I used to mark the claim as true. Please validate the line reasoning.

Answer 1

Yes, it is correct. Only you have to be carefull about the order of the sentences:

Since $Y \subseteq X$ then for each $Z$ we have: $$ Y \cup Z \subseteq X \cup Z.$$

so $$ (X \cup Z)^{\complement} \subseteq (Y \cup Z)^{\complement}$$ Now apply de Morgan and thus conclusion.

Answer 2

Since $Y \subseteq X, X^c \subseteq Y^c.$
Thus for all A, $X^c \cap A \subseteq Y^c \cap A.$

Answer 3

Option:

Given : $Y\subset X$, implies

$X^c\subset Y^c.$

Since:

$y \in Y$ implies $y \in X$, then

$y \not\in X$ implies $y \not \in Y$, or

$y \in X^c$ implies $y \in Y^c$, or

$X^c \subset Y^c.$

Let $A$ be any set, then:

$A \cap X^c \subset A \cap Y^c$ (Why?).