Answer :
(a) 141 people, (b) 18 people, (c) 45 people, (d) 59 people owned neither a Blu-ray player nor a microwave oven.
To solve this problem, let's use the principle of inclusion-exclusion:
(a) To find the number of people who owned either a Blu-ray player or a microwave oven, we add the number of people who owned a Blu-ray player to the number of people who owned a microwave oven and then subtract the number of people who owned both to avoid double-counting.
[tex]\[ \text{(a)} = \text{Blu-ray player} + \text{Microwave oven} - \text{Both} \][/tex]
[tex]\[ \text{(a)} = 96 + 123 - 78 = 141 \][/tex]
(b) To find the number of people who owned a Blu-ray player but not a microwave oven, we subtract the number of people who owned both from the total number of Blu-ray player owners.
[tex]\[ \text{(b)} = \text{Blu-ray player} - \text{Both} \][/tex]
[tex]\[ \text{(b)} = 96 - 78 = 18 \][/tex]
(c) To find the number of people who owned a microwave oven but not a Blu-ray player, we subtract the number of people who owned both from the total number of microwave oven owners.
[tex]\[ \text{(c)} = \text{Microwave oven} - \text{Both} \][/tex]
[tex]\[ \text{(c)} = 123 - 78 = 45 \][/tex]
(d) To find the number of people who owned neither a Blu-ray player nor a microwave oven, we subtract the total number of people who owned either or both devices from the total number of people surveyed.
[tex]\[ \text{(d)} = \text{Total} - \text{(a)} \][/tex]
[tex]\[ \text{(d)} = 200 - 141 = 59 \][/tex]
So, the answers are:
(a) 141 people
(b) 18 people
(c) 45 people
(d) 59 people
Using the inclusion-exclusion principle, the number of people who owned a Blu-ray player or a microwave oven is 141, 18 owned only a Blu-ray player, 45 owned only a microwave oven, and 59 owned neither.
The question involves applying principles of set theory to a real-world problem related to ownership of Blu-ray players and microwave ovens. To solve this, we utilize the inclusion-exclusion principle. Let's denote the number of people who own a Blu-ray player as B and the number of people who own a microwave oven as M. According to the question:
B (Blu-ray owners) = 96
M (Microwave owners) = 123
B \\cap M (Both Blu-ray and microwave owners) = 78
To answer the following:
(a) a Blu-ray player or a microwave oven: We use the formula Total = B + M - B \\cap M.
Thus, Total = 96 + 123 - 78 = 141 people.
(b) a Blu-ray player but not a microwave oven: This is calculated as B - B \\cap M.
Thus, we have 96 - 78 = 18 people.
(c) a microwave oven but not a Blu-ray player: This is M - B \\cap M.
Hence, 123 - 78 = 45 people.
(d) neither a Blu-ray player nor a microwave oven: This is the total number surveyed minus those that own either one, so 200 - 141 = 59 people.
