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), and set \( Q \) contains odd integers. Even and odd integers are mutually exclusive, so \( P \cap Q=\varnothing \) (no overlap between \( P \) and \( Q \)).

Step2: Analyze Set R with P and Q

• Set \( R \) (integers divisible by 3) has elements that can be even (e.g., 6, which is in \( P \) and \( R \)) or odd (e.g., 9, which is in \( Q \) and \( R \)). So \( R \) overlaps with both \( P \) and \( Q \), but \( P \) and \( Q \) do not overlap with each other.

Step3: Match with Diagrams

• The first diagram: \( P \) and \( Q \) are separate, and \( R \) overlaps with both \( P \) and \( Q \) (since \( R \) has even and odd multiples of 3), which fits.
• The second diagram: \( P \) and \( Q \) overlap, which is wrong (as even and odd are disjoint).
• The third diagram: \( Q \) and \( R \) do not overlap, which is wrong (since odd multiples of 3 exist, e.g., 9 is in \( Q \) and \( R \)).

Answer:

The first Venn diagram (with \( P \) and \( Q \) separate, \( R \) overlapping both \( P \) and \( Q \))