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Is this phrase true?

$A⊆B ∧ C⊆D \Rightarrow A∩C ⊆ B∩D$

In first look there is no connection between $A⊆B$ and $C⊆D$ but $A∩C$ in the worst situation is $∅$ and $∅$ is subset of all sets, so can we say if $A⊆B$$C⊆D$, $A∩C$ is subset of $B∩D$?

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2 Answers 2

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We want to show that whenever we find something in $A \cap C$ it also belongs to $B \cap D$ that is $A \cap C \subseteq B \cap D$

Consider $x \in A \cap C$. You then know that $x$ belongs to $A$ and $C$.

Because $x \in A , x \in B$ because $A \subseteq B $

Similarly for $x \in C$

Can you finish from here?

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$A\cap C \subseteq B\cap C \subseteq B\cap D$.

Alternatively,

$A\cap C \subseteq A\cap D \subseteq B\cap D$.

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