Question

which diagram represents the following sets? p = {x | x is an integer divisible by 2}; q = {x | x is an odd integer}; r = {x | x is an integer divisible by 3}

Explanation:

Step1: Analyze set P and Q

Set \( P \) contains even integers (divisible by 2), set \( Q \) contains odd integers. An integer cannot be both even and odd, so \( P \cap Q=\varnothing \) (no overlap between \( P \) and \( Q \)).

Step2: Analyze set P and R

Set \( P \) (even integers) and set \( R \) (integers divisible by 3) can have common elements (e.g., 6, 12, etc.), so \( P \cap R eq\varnothing \) (they overlap).

Step3: Analyze set Q and R

Set \( Q \) (odd integers) and set \( R \) (integers divisible by 3) can have common elements (e.g., 3, 9, etc.), but from the diagrams, we first check \( P \) and \( Q \) overlap. Since \( P \) and \( Q \) have no overlap, we eliminate the diagram where \( P \) and \( Q \) overlap (the middle one). Then, check \( P \) and \( R \) overlap, and \( Q \) and \( P \) no overlap. The third diagram has \( P \) and \( R \) overlapping, \( Q \) and \( P \) not overlapping, which matches. Wait, no—wait, first diagram: \( P \) and \( Q \) are separate, \( P \) and \( R \) overlap, \( Q \) and \( R \) overlap? Wait, no, let's re - check. Wait, \( P \) (even) and \( Q \) (odd) must be disjoint. So in the diagrams:

- Middle diagram: \( P \) and \( Q \) overlap (has a common region), which is impossible, so eliminate.

- Third diagram: \( P \) and \( Q \) are separate (no overlap), \( P \) and \( R \) overlap (good, since even and divisible by 3 exist), \( Q \) and \( R \): in the third diagram, \( Q \) and \( R \) are separate? Wait, no, the first diagram: \( P \) (blue), \( Q \) (red) are separate, \( P \) and \( R \) (green) overlap, \( Q \) and \( R \) overlap. But \( Q \) is odd, \( R \) is divisible by 3: odd and divisible by 3 (like 3, 9) are in \( Q \cap R \), so \( Q \) and \( R \) should overlap. Wait, but \( P \) and \( Q \) must not overlap. The third diagram: \( Q \) and \( P \) are separate, \( P \) and \( R \) overlap, \( Q \) and \( R \) are separate? No, that's wrong. Wait, no—wait the first diagram: \( P \) (blue) and \( Q \) (red) are two circles with no overlap (good, since even and odd are disjoint), \( P \) and \( R \) (green) overlap (good, even and divisible by 3), \( Q \) and \( R \) (green) overlap (good, odd and divisible by 3). Wait, but the third diagram: \( Q \) and \( R \) are separate? No, the third diagram's \( Q \) is a red circle, \( R \) is a green circle, and they don't overlap. But \( Q \) (odd) and \( R \) (divisible by 3) should overlap (e.g., 3 is in both \( Q \) and \( R \)). Wait, I made a mistake. Wait, the first diagram: \( P \) (blue) and \( Q \) (red) are separate (no overlap, good), \( P \) and \( R \) (green) overlap, \( Q \) and \( R \) (green) overlap. The middle diagram has \( P \) and \( Q \) overlapping (bad, since even and odd can't be same). The third diagram: \( P \) and \( R \) overlap, \( Q \) and \( P \) no overlap, but \( Q \) and \( R \) no overlap (bad, because 3 is in \( Q \) and \( R \)). Wait, so the first diagram: \( P \) (blue) and \( Q \) (red) are disjoint, \( P \cap R\) (overlap), \( Q \cap R\) (overlap). Let's check the sets again:

\( P=\{x|x \text{ is even integer}\} \), \( Q = \{x|x\text{ is odd integer}\} \), so \( P \) and \( Q \) are disjoint (no common elements). \( R=\{x|x\text{ is divisible by }3\} \). So \( P \) and \( R \) can have common elements (even multiples of 3: 6, 12,...), \( Q \) and \( R \) can have common elements (odd multiples of 3: 3, 9,...). So the Venn diagram should have:

- \( P \) and \( Q \): no overlap.

- \( P \) and \( R \): overlap.

- \( Q \) and \( R \): overlap.

Looking at the three diagrams:

1. First diagram: \( P \) (blue) and \( Q \) (red) are separate (no overlap), \( P \) and \( R \) (green) overlap, \( Q \) and \( R \) (green) overlap. This matches.

2. Middle diagram: \( P \) and \( Q \) overlap (bad, since even and odd can't be same), so eliminate.

3. Third diagram: \( Q \) and \( R \) are separate (bad, since 3 is in both \( Q \) and \( R \)), so eliminate.

Wait, but the third diagram's \( Q \) and \( R \) are separate? Let me re - look. The third diagram: \( P \) (blue) and \( R \) (green) overlap, \( Q \) (red) is separate from both \( P \) and \( R \)? No, that's not right. So the first diagram is correct? Wait, no, the user's third diagram: the blue (P) and green (R) overlap, red (Q) is separate from blue (P) (good, since P and Q are disjoint) but red (Q) is also separate from green (R)? But Q and R should overlap (e.g., 3 is in Q (odd) and R (divisible by 3)). So I think I made a mistake. Wait, maybe the first diagram: \( P \) (blue) and \( Q \) (red) are separate, \( P \) and \( R \) overlap, \( Q \) and \( R \) overlap. The middle diagram has P and Q overlapping (invalid). The third diagram: P and R overlap, Q and P no overlap, Q and R no overlap (invalid, because Q and R should overlap). Wait, maybe the problem is that in the third diagram, Q and R are not overlapping, but in reality, Q (odd) and R (divisible by 3) can overlap. But maybe the options are as follows: the first diagram: P (blue), Q (red) are separate, P and R overlap, Q and R overlap. Middle: P and Q overlap. Third: P and R overlap, Q and P no overlap, Q and R no overlap. So the correct diagram is the first one? Wait, no, the user's third diagram: the red circle (Q) is next to the blue (P) and green (R), with P and R overlapping, Q and P not, Q and R not. But Q and R should overlap. Wait, maybe I misread the diagrams. Let's re - express:

Set \( P \): even integers (so all elements are even).

Set \( Q \): odd integers (all elements are odd).

So \( P \) and \( Q \) are disjoint (no elements in common). So in Venn diagram, circles for \( P \) and \( Q \) should not overlap.

Set \( P \) (even) and \( R \) (divisible by 3): have common elements (6, 12, etc.), so \( P \) and \( R \) overlap.

Set \( Q \) (odd) and \( R \) (divisible by 3): have common elements (3, 9, etc.), so \( Q \) and \( R \) overlap.

So the Venn diagram must have:

- \( P \) and \( Q \): no overlap.

- \( P \) and \( R \): overlap.

- \( Q \) and \( R \): overlap.

Looking at the three diagrams:

1. First diagram: \( P \) (blue) and \( Q \) (red) are separate (no overlap), \( P \) and \( R \) (green) overlap, \( Q \) and \( R \) (green) overlap. This fits.

2. Middle diagram: \( P \) and \( Q \) overlap (has a common region), which is impossible, so eliminate.

3. Third diagram: \( P \) and \( R \) overlap, \( Q \) and \( P \) no overlap, but \( Q \) and \( R \) no overlap (since \( Q \) is separate from \( R \)), which is wrong because \( Q \) and \( R \) should overlap.

So the first diagram (the left - most one) is correct? Wait, but the user's third diagram: the blue (P) and green (R) overlap, red (Q) is separate from blue (P) (good) but red (Q) is also separate from green (R). So that's incorrect. So the first diagram is correct.

Answer:

The left - most diagram (the first one among the three)