Which of the following correctly shades the union of sets A and B?

Question

Which of the following correctly shades the union of sets A and B?

BlackTom AI · Accepted Answer

The union A ∪ B consists of all elements that are in A or in B or in both, so the correct shading must include every point inside either circle (including the overlap).

1. Relevant knowledge points and concepts
- Definition: A ∪ B = {x | x ∈ A or x ∈ B (or both)}.  
- Venn diagram shading: shading a region means those elements are included in the set; the overlap region (intersection) is part of both A and B and therefore must be included in the union.

2. Step-by-step reasoning
- Identify all regions in a two-set Venn diagram: (i) A only, (ii) B only, (iii) A ∩ B (the overlap), (iv) outside both (the universal set S but not in A or B).  
- A ∪ B includes regions (i), (ii), and (iii). It excludes only region (iv).  
- Therefore the correct diagram shades the entire area of both circles, including their middle overlap.

3. (Logical) intermediate steps
- Check each option against the requirement "shade A only, B only, and overlap":  
  - If an option leaves the overlap unshaded, it fails because those overlap elements are in both A and B and must be included.  
  - If an option shades only the overlap, it represents A ∩ B, not the union.  
  - If an option shades only A (or only B), it represents A (or B) rather than A ∪ B.

4. Why the correct answer is correct
- The top diagram (first option) shades both whole circles including their overlap, which exactly matches the definition of A ∪ B (elements in A or B or both).

5. Why the other options are incorrect
- Second option: circles shaded but the overlap left unshaded — this is the symmetric difference (A Δ B), not the union.  
- Third option: only the overlap shaded — this is A ∩ B.  
- Fourth option: only A shaded excluding the overlap — this is A \ B.

Conclusion: The top (first) diagram correctly represents A ∪ B; therefore the correct choice is the first option.

Which of the following correctly shades the union of sets A and B?单项选择题

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