To prove that $A \subseteq B$ if and only if $B^c \subseteq A^c$, we must demonstrate both directions of the equivalence:

1. If $A \subseteq B$, then $B^c \subseteq A^c$.
2. If $B^c \subseteq A^c$, then $A \subseteq B$.

Here is the detailed proof:

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1. If $A \subseteq B$, then $B^c \subseteq A^c$:

• Suppose $A \subseteq B$. This means that for every element $x \in U$: $x \in A \implies x \in B.$
• The contrapositive of this implication is: $x \notin B \implies x \notin A.$
• In terms of complements, this means: $x \in B^c \implies x \in A^c.$
• Therefore, every element of $B^c$ is also an element of $A^c$, so: $B^c \subseteq A^c.$

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2. If $B^c \subseteq A^c$, then $A \subseteq B$:

• Suppose $B^c \subseteq A^c$. This means that for every element $x \in U$: $x \in B^c \implies x \in A^c.$
• Equivalently: $x \notin B \implies x \notin A.$
• Taking the contrapositive: $x \in A \implies x \in B.$
• Therefore, every element of $A$ is also an element of $B$, so: $A \subseteq B.$

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Conclusion:

Since both directions have been proven, we conclude that: $A \subseteq B \iff B^c \subseteq A^c.$

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Would you like more examples or details? Here are some questions to deepen understanding:

1. How does this proof rely on the contrapositive of logical implications?
2. Can you provide a concrete example where $A \subseteq B$ and verify that $B^c \subseteq A^c$?
3. How do these subset relationships work in the context of Venn diagrams?
4. What happens if $A$ and $B$ are disjoint sets?
5. How is the concept of complements used in Boolean algebra or set theory?

Tip: Always try visualizing set relationships using Venn diagrams to solidify understanding.