Answer
- **(a)** Place \(A\) inside \(C\) and make sure \(B\) intersects \(C\) but does not touch \(A\).
- **(b)** Place both \(A\) and \(C\) inside \(B\), ensuring \(A\cap C\neq\varnothing\).
- **(c)** \(A\) must be empty to satisfy \(A\subset B\) yet \(A\cap B=\varnothing\). Then put a nonempty \(C\) inside \(B\).
Work Step by Step
Here is one way to reason out each case and sketch suitable Venn diagrams.
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## General Strategy
You have three sets \(A, B, C\). A “Venn diagram” typically places each set as a circle/oval so that all possible intersections can be visualized. However, sometimes the conditions force certain intersections to be empty or one set to be contained within another. In those situations, it may be clearer to draw one circle entirely inside another or to indicate that one set is empty, etc.
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## (a) Conditions
1. \(A \cap B = \varnothing\).
2. \(A \subset C\).
3. \(C \cap B \neq \varnothing\).
### Reasoning
- **\(A \subset C\)** means \(A\) is entirely inside \(C\).
- **\(A \cap B = \varnothing\)** means \(A\) and \(B\) do not overlap at all.
- Meanwhile, **\(C \cap B \neq \varnothing\)** means \(C\) and \(B\) *do* overlap somewhere, but that overlapping region cannot include \(A\) (since \(A\) is disjoint from \(B\)).
### Sketch
1. Draw a circle for \(C\).
2. Place \(A\) entirely *inside* \(C\). (A smaller circle within \(C\).)
3. Draw \(B\) so that it overlaps part of \(C\) **but** *not* the region where \(A\) lies.
The overlap between \(C\) and \(B\) exists (so \(C \cap B \neq \varnothing\)), but \(A\) is in a portion of \(C\) that does *not* intersect \(B\).
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## (b) Conditions
1. \(A \subset B\).
2. \(C \subset B\).
3. \(A \cap C \neq \varnothing\).
### Reasoning
- Both \(A\) and \(C\) must lie entirely within \(B\).
- Additionally, \(A\) and \(C\) must overlap each other in some region (so their intersection is nonempty).
### Sketch
1. Draw a large circle for \(B\).
2. Inside it, draw two circles, one for \(A\) and one for \(C\).
3. Make sure \(A\) and \(C\) *overlap* within \(B\).
Everything is contained in \(B\), and \(A\) meets \(C\) in a nonempty region.
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## (c) Conditions
1. \(A \cap B = \varnothing\).
2. \(B \cap C \neq \varnothing\).
3. \(A \cap C = \varnothing\).
4. \(A \subset B\).
5. \(C \subset B\).
### Reasoning
- We want \(A\subset B\) but \(A\cap B=\varnothing\). The *only* way for a nonempty set \(A\) to be a subset of \(B\) and yet have no elements in common is impossible.
- Therefore, **\(A\) must be empty**. (An empty set is technically a subset of *every* set, and of course it is disjoint from every set.)
- Meanwhile, \(C\subset B\) but \(C\) must *not* be empty, because \(B \cap C\neq \varnothing\).
- Also, \(A \cap C=\varnothing\) holds trivially if \(A\) is empty.
### Sketch
- Draw \(B\) as a circle.
- Inside \(B\), place \(C\) as a nonempty region.
- \(A\) is the empty set, so it does not appear as a region at all (or you can label an empty circle off to the side to emphasize it has no points).
That satisfies all the conditions: \(A\subset B\) vacuously (empty is a subset of everything), \(A\cap B=\varnothing\), \(C\subset B\) and \(C\neq\varnothing\), so \(B\cap C\neq\varnothing\), and trivially \(A\cap C=\varnothing\).
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