Question

an animal sanctuary houses a variety of animals. to understand the workload of the sanctuarys veterinarians, the director looked at which types of animals are treated or not treated by various doctors. the venn diagram shows this information for three of the doctors.
(a) select all the doctors who do not treat zebras.
☐ dr. howard ☐ dr. bell ☐ dr. martinez
(b) how many types of animals are treated by dr. howard or dr. martinez (or both)?
(c) which types of animals are treated by neither dr. howard nor dr. bell nor dr. martinez? choose all that apply.
☐ birds ☐ cats ☐ elks ☐ frogs ☐ goats
☐ horses ☐ koalas ☐ llamas ☐ otters ☐ rabbits
☐ snakes ☐ turtles ☐ zebras

Explanation:

Part (a)

To determine which doctors do not treat zebras, we look at the Venn diagram. Zebras are in the intersection of Dr. Howard and Dr. Bell's circles. So Dr. Martinez's circle does not include zebras, and we check the other doctors: Dr. Howard treats zebras (since zebras are in his circle), Dr. Bell treats zebras (in his circle), Dr. Martinez does not. Wait, no—wait, the Venn diagram: "Treated by Dr. Howard" circle has Llamas, Otters, Zebras, Rabbits, Elks? Wait, no, let's parse the Venn:

• Dr. Howard's circle: Llamas, Otters, Zebras, Rabbits, Elks (wait, the overlapping regions: Elks is in Howard, Bell, Martinez? Wait, the labels:

• Dr. Howard: Llamas, Otters, Zebras (and overlapping with Bell: Zebras? Wait, the Venn has three circles: Dr. Howard, Dr. Bell, Dr. Martinez.

• Zebras are in the intersection of Dr. Howard and Dr. Bell (since it's in the "Treated by Dr. Howard" and "Treated by Dr. Bell" overlap, not in Dr. Martinez's circle). So Dr. Martinez does not treat zebras. Now check Dr. Bell: Zebras are in his circle (overlap with Howard), so Dr. Bell treats zebras. Dr. Howard treats zebras. So the doctor who does not treat zebras is Dr. Martinez. Wait, but the options are Dr. Howard, Dr. Bell, Dr. Martinez. So we need to select the ones who do NOT treat zebras. So Dr. Martinez (since zebras are not in his circle), and wait, does Dr. Bell treat zebras? Zebras are in the overlap of Howard and Bell, so yes, Dr. Bell treats zebras. Dr. Howard treats zebras. So only Dr. Martinez does not treat zebras? Wait, no—wait, maybe I misread. Let's re-express the Venn:

• Dr. Howard's circle (only his): Llamas, Otters

• Dr. Howard and Bell (not Martinez): Zebras

• Dr. Howard and Martinez (not Bell): Rabbits

• All three (Howard, Bell, Martinez): Elks

• Dr. Bell's circle (only his): Birds, Turtles, Koalas

• Dr. Bell and Martinez (not Howard): Frogs

• Dr. Martinez's circle (only his): Horses

• Outside all three: Snakes, Goats, Cats

So Zebras are in Howard and Bell (not Martinez). So Dr. Howard treats zebras (yes, in his circle), Dr. Bell treats zebras (yes, in his circle), Dr. Martinez does not (no, not in his circle). So the doctors who do not treat zebras are Dr. Martinez. Wait, but the question is "Select all the doctors who do not treat zebras." So only Dr. Martinez? Wait, maybe I made a mistake. Wait, the Venn: "Treated by Dr. Howard" includes Zebras, "Treated by Dr. Bell" includes Zebras (in their overlap), and Dr. Martinez's circle does not include Zebras. So Dr. Howard: treats zebras (yes), Dr. Bell: treats zebras (yes), Dr. Martinez: does not (no). So the answer for (a) is Dr. Martinez. Wait, but maybe I messed up. Let's confirm: Zebras are in the intersection of Howard and Bell, so both Howard and Bell treat zebras. Martinez's circle has Rabbits (Howard and Martinez), Elks (all three), Frogs (Bell and Martinez), Horses (only Martinez), and outside: Snakes, Goats, Cats. So Zebras are not in Martinez's circle, so Martinez does not treat zebras. So select Dr. Martinez.

Wait, but the options are checkboxes: Dr. Howard (check? No, he treats zebras), Dr. Bell (check? No, he treats zebras), Dr. Martinez (check? Yes, he does not treat zebras). So the answer for (a) is Dr. Martinez.

Part (b)

To find the number of animals treated by Dr. Howard or Dr. Martinez (or both), we use the principle of inclusion - exclusion: \( n(A \cup B)=n(A)+n(B)-n(A \cap B) \), where \( A \) is animals treated by Dr. Howard, \( B \) by Dr. Martinez.

Step 1: Count animals in Dr. Howard's circle (including overlaps)

• Only Howard: Llamas, Otters (2)
• Howard & Bell (not Martinez): Zebras (1)
• Howard & Martinez (not Bell): Rabbits (1)
• All three (Howard, Bell, Martinez): Elks (1)
• Total for Dr. Howard: \( 2 + 1+1 + 1=5 \)

Step 2: Count animals in Dr. Martinez's circle (including overlaps)

• Only Martinez: Horses (1)
• Howard & Martinez (not Bell): Rabbits (1)
• Bell & Martinez (not Howard): Frogs (1)
• All three (Howard, Bell, Martinez): Elks (1)
• Total for Dr. Martinez: \( 1+1 + 1+1 = 4 \)

Step 3: Count animals in both (Howard and Martinez)

• Howard & Martinez (not Bell): Rabbits (1)
• All three: Elks (1)
• Total overlap: \( 1 + 1=2 \)

Step 4: Apply inclusion - exclusion

\( n(\text{Howard or Martinez})=n(\text{Howard})+n(\text{Martinez})-n(\text{Howard and Martinez}) \)
\( = 5+4 - 2=7 \)

Wait, let's list all animals in Howard or Martinez:

• From Howard: Llamas, Otters, Zebras, Rabbits, Elks

• From Martinez (not already in Howard): Horses, Frogs

So list them: Llamas, Otters, Zebras, Rabbits, Elks, Horses, Frogs. That's 7 animals.

Animals treated by neither Dr. Howard, Dr. Bell, nor Dr. Martinez are outside all three circles. From the Venn, the animals outside are Snakes, Goats, Cats. So we check the options:

• Birds: in Dr. Bell's circle (no)

• Cats: outside (yes)

• Elks: in all three (no)

• Frogs: in Bell and Martinez (no)

• Goats: outside (yes)

• Horses: in Dr. Martinez (no)

• Koalas: in Dr. Bell (no)

• Llamas: in Dr. Howard (no)

• Otters: in Dr. Howard (no)

• Rabbits: in Howard and Martinez (no)

• Snakes: outside (yes)

• Turtles: in Dr. Bell (no)

• Zebras: in Howard and Bell (no)

So the animals are Cats, Goats, Snakes.

Answer:

7

Part (c)