Question

question 6
1 pts
in the venn diagram, what region(s) make up \\(\overline{a}\\)?
\\(\bigcirc\\) ii, iii, and iv
\\(\bigcirc\\) i and iii
\\(\bigcirc\\) i, ii, and iv
\\(\bigcirc\\) i and iv
\\(\bigcirc\\) ii
\\(\bigcirc\\) i, ii, and iii
\\(\bigcirc\\) iii and iv
\\(\bigcirc\\) iv

Explanation:

Step1: Recall the definition of complement

The complement of a set (or event) \( \overline{A} \) (read as "A bar") consists of all elements (or outcomes) that are not in set \( A \).

Step2: Identify regions in Venn Diagram

- Region I: Inside Event A (only A, not overlapping with B)
- Region II: Overlap of A and B (in both A and B)
- Region III: Inside Event B (only B, not overlapping with A)
- Region IV: Outside both A and B

So, the regions not in A are Region III (only B) and Region IV (outside both A and B), and also Region II? Wait, no. Wait, Event A includes Region I (only A) and Region II (A and B). So \( \overline{A} \) is all regions not in A. So regions not in A are:
- Region III: only B (not in A)
- Region IV: outside both (not in A)
- Wait, Region II is in A (since it's the overlap of A and B). Wait, no: Event A is the yellow circle, which includes Region I (yellow only) and Region II (green, overlap). So \( \overline{A} \) is everything not in the yellow circle (Event A). So the yellow circle is I and II. So not in A: III (blue only) and IV (outside both). Wait, but the options include "III and IV"? Wait, let's check the options again.

Wait the options are:
- II, III, and IV
- I and III
- I, II, and IV
- I and IV
- II
- I, II, and III
- III and IV
- IV

Wait, let's re-express:

Event A (yellow circle) has regions: I (yellow only) and II (green, overlap with B). So \( \overline{A} \) is all regions not in A: that is, regions not in I or II. So:

- Region III: blue only (not in A)
- Region IV: outside both (not in A)

Wait, but Region II is in A (since it's part of the yellow circle? Wait no, the yellow circle is Event A, the blue is Event B. The green region (II) is the intersection of A and B, so it is in A (because it's inside the yellow circle). So \( \overline{A} \) is regions not in A: so III (blue only, not in A) and IV (outside both, not in A). Wait, but the first option is "II, III, and IV" – that can't be, because II is in A. Wait maybe I made a mistake. Wait, maybe the Venn Diagram: Event A is yellow (I and II), Event B is blue (II and III), and IV is outside both. So \( \overline{A} \) is all elements not in A. So elements not in A are those not in I or II. So:

- Region III: in B, not in A (since II is in A, III is only B)
- Region IV: not in A or B
- Wait, Region II is in A (because it's in the yellow circle, Event A). So \( \overline{A} \) is III and IV? Wait, but the option "III and IV" is there. Wait, but let's check again.

Wait, maybe the initial analysis is wrong. Let's think again:

The complement of A, \( \overline{A} \), is the set of all outcomes not in A. So in the Venn Diagram:

- A is the yellow circle: includes I (yellow only) and II (green, overlap with B). So \( \overline{A} \) is everything not in the yellow circle. So:

- Region III: blue circle, not overlapping with yellow (so not in A)
- Region IV: outside both circles (not in A)
- Region II: is in the yellow circle? Wait, the yellow circle is Event A, so the yellow circle includes I (yellow) and II (green, which is the overlap). So II is in A. So \( \overline{A} \) is III and IV. Wait, but the option "III and IV" is present. Wait, but let's check the options again.

Wait the options are:

1. II, III, and IV

2. I and III

3. I, II, and IV

4. I and IV

5. II

6. I, II, and III

7. III and IV

8. IV

So according to the definition, \( \overline{A} \) is regions not in A. A is I (only A) and II (A and B). So not in A: III (only B) and IV (outside both). So the correct option is "III and IV". Wait, but let's confirm with the Venn Diagram structure.

Alternatively, maybe the green region (II) is not in A? Wait, no, the yellow circle is Event A, so the area inside Event A is I (yellow) and II (green, intersection). So II is in A. Therefore, \( \overline{A} \) is III (blue only) and IV (outside). So the answer is "III and IV".

Wait, but let's check the first option: "II, III, and IV" – that would mean II is in \( \overline{A} \), but II is in A (since it's in the yellow circle). So that's wrong. So the correct regions are III (not in A) and IV (not in A). So the option "III and IV" is correct.

Answer:

III and IV