MHB Questions about sets and subsets

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The discussion centers on proving set relationships involving subsets X and Y of a universal set U. Specifically, it addresses the implication that if X is a subset of Y, then the complement of Y is a subset of the complement of X. A suggested approach includes using a Venn diagram to visualize the relationship, although a textual explanation suffices. The explanation provided clarifies that any element not in Y is also not in X, thereby confirming the subset relationship for their complements. This proof emphasizes the logical connections between subsets and their complements in set theory.
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Hi, the question goes as follows:

Given two subsets X and Y of a universal set U, prove that: (refer to picture)

I'm having particular trouble on D and E, if anyone can clear this up or provide some clarification for me that would be appreciated! I know a venn diagram might not be possible here so just an explanation is ok

Thank you!​
 

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shle said:
Hi, the question goes as follows:

Given two subsets X and Y of a universal set U, prove that: (refer to picture)

I'm having particular trouble on D and E, if anyone can clear this up or provide some clarification for me that would be appreciated! I know a venn diagram might not be possible here so just an explanation is ok

Thank you!​

d) Show that \( X\subseteq Y\) implies \(\overline{Y} \subseteq \overline{X}\)

Other than the Venn diagram where the region representing \(X\) is inside that representin \(Y\) so that the coplement of \(Y\) is contained within the complement of \(X\) (You are strongly recomended to draw the diagram), Consider any \(z \not\in Y\), then it is in \(\overline{Y}\), but because \( X\subseteq Y\) it is also not in \(X\), so is in \(\overline{X}\), which proves the result.

CB
 
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