Question

a survey of 1200 people in a certain city indicates that 860 own microwave ovens, 730 own dvd players, and 570 own microwave ovens and dvd players.
(a) how many people in the survey own either a microwave oven or a dvd player?
(b) how many own neither a microwave oven nor a dvd player?
(c) how many own a microwave oven and do not own a dvd player?

(a) \\(\square\\) people in the survey own either a microwave oven or a dvd player.
(b) \\(\square\\) people in the survey own neither a microwave oven nor a dvd player.
(c) \\(\square\\) people in the survey own a microwave oven and do not own a dvd player.

Explanation:

Step1: Define sets and values

Let \( M = \text{microwave owners} = 860 \), \( D = \text{DVD owners} = 730 \), \( M \cap D = 570 \), total \( N = 1200 \).

Step2: Calculate (A) using inclusion-exclusion

\( |M \cup D| = |M| + |D| - |M \cap D| = 860 + 730 - 570 = 1020 \).

Step3: Calculate (B) as total minus union

\( N - |M \cup D| = 1200 - 1020 = 180 \).

Step4: Calculate (C) as microwave minus intersection

\( |M| - |M \cap D| = 860 - 570 = 290 \).

Answer:

(A) 1020
(B) 180
(C) 290