Question

and venn diagrams
the venn diagram shows the number of customers who have purchased
different types of pets from a pet store, where c represents customers who
have purchased cats, d represents customers who have purchased dogs,
and f represents customers who have purchased fish.
how many people are in the set ( c cap d )?
6
4
36
38

Explanation:

Step1: Understand \( C \cap D \)

\( C \cap D \) is the intersection of sets \( C \) (cats) and \( D \) (dogs), including all regions common to both \( C \) and \( D \).

Step2: Identify regions in \( C \cap D \)

In the Venn diagram, the regions common to \( C \) and \( D \) are the part where only \( C \) and \( D \) overlap (value 3) and the part where \( C \), \( D \), and \( F \) (fish) overlap (value 1).

Step3: Calculate the total

Add these two values: \( 3 + 1 = 4 \)? Wait, no—wait, wait, the region for \( C \cap D \) without \( F \) is 3, and with \( F \) is 1? Wait, no, let's re - check. Wait, the intersection of \( C \) and \( D \) includes the area that is in both \( C \) and \( D \), regardless of \( F \). So the regions are the yellow (3) and the purple (1) that is in both \( C \) and \( D \). Wait, no, actually, \( C \cap D \) is the set of elements in both \( C \) and \( D \). So in the Venn diagram, the parts that are in both \( C \) and \( D \) are the 3 (only \( C \) and \( D \)) and the 1 (all three). Wait, but maybe I misread. Wait, the problem is asking for \( C \cap D \), which is the number of people who bought both cats and dogs. So we need to sum the numbers in the overlapping regions of \( C \) and \( D \). The overlapping regions of \( C \) and \( D \) are the 3 (only cats and dogs) and the 1 (cats, dogs, and fish). So \( 3+1 = 4 \)? Wait, no, wait the options include 4? Wait, the options are 6, 4, 36, 38. Wait, maybe I made a mistake. Wait, maybe the region for \( C \cap D \) is 3 (only \( C \) and \( D \)) plus 1 (all three)? Wait, no, let's look again. The Venn diagram: \( C \) has 15 (only \( C \)), 3 ( \( C \) and \( D \) only), 2 ( \( C \) and \( F \) only), 1 (all three). \( D \) has 21 (only \( D \)), 3 ( \( C \) and \( D \) only), 0 ( \( D \) and \( F \) only), 1 (all three). So \( C \cap D \) is the set of elements in both \( C \) and \( D \), so that's 3 ( \( C \cap D \) only) plus 1 ( \( C \cap D \cap F \))? Wait, no, \( C \cap D \) is all elements that are in both \( C \) and \( D \), regardless of \( F \). So \( n(C \cap D)=3 + 1=4 \)? But wait, the first option is 6? Wait, maybe I added wrong. Wait, 3 ( \( C \) and \( D \) only) and 3? No, wait the diagram: the part between \( C \) and \( D \) (not including \( F \)) is 3, and the part between \( C \), \( D \), and \( F \) is 1. Wait, no, maybe the 3 is the \( C \cap D \) only, and the 1 is \( C \cap D \cap F \). So \( C \cap D=(C \cap D \cap \overline{F})+(C \cap D \cap F)=3 + 1 = 4 \). So the answer should be 4.

Answer:

4