Subjects set theory

Complement Intersection 25A281

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Complement Intersection 25A281


1. **State the problem:** We need to shade the complement of the intersection of sets $C$ and $A$, denoted as $(C \cap A)'$. 2. **Recall definitions:** - The intersection $C \cap A$ is the set of elements common to both $C$ and $A$. - The complement $(C \cap A)'$ is everything in the universal set $U$ that is *not* in $C \cap A$. 3. **Express the complement using set operations:** $$(C \cap A)' = U - (C \cap A)$$ 4. **Use De Morgan's Law:** De Morgan's Law states: $$ (C \cap A)' = C' \cup A' $$ This means the complement of the intersection is the union of the complements. 5. **Interpretation for shading:** - Shade all areas outside $C$ (i.e., $C'$) and all areas outside $A$ (i.e., $A'$). - This includes everything except the overlapping region of $C$ and $A$. 6. **Summary:** To shade $(C \cap A)'$, shade all parts of the universal set except the region where $C$ and $A$ overlap. **Final answer:** $$(C \cap A)' = C' \cup A'$$ This means shade all areas outside the intersection of $C$ and $A$.