150 men live in a certain town:
85 are married
70 have a cell phone
75 own a car
55 are married and have a cell phone
35 have a cellphone and a car
40 are married and have car
30 are married, have a car and have a cellphone. How many men are single and do not have a car? How many men are single and do not have either a car or a cellphone?
I am trying to set this up using sets and/or venn diagram. can anyone assist me?
Draw your Venn diagram of three intersecting circles.
30 in the center part
Since 55 are married and have a cell phone, that leaves 25 in the unlabeled intersection of married and cell.
Similarly, 5 in the cell-car intersection not already labeled.
25 in the unlabeled part of married-car intersection.
Now label the still empty parts by adding up all three numbers entered for each large circle, and subtracting from the total for the large circle. That makes the numbers in the still empty parts
20 married only
10 cell only
30 car only
85+70+75 - (55+35+40) + 30 = 130, so there are 20 not in any circle: single, no car, no cell.
single & no car = 20+10 = 30
the 10 come from those not married, with no car but with cell.
20 are single, no car, no cell: U-(M&cell&car)
Yes, I can assist you in setting up the problem using a Venn diagram.
Let's represent the sets as follows:
M = Men in the town
C = Men who have a car
P = Men who have a cellphone
S = Single men
We are given the following information:
M = 150
M(C) = 75 (men who have a car)
M(P) = 70 (men who have a cellphone)
M(M∩P) = 55 (men who are married and have a cellphone)
M(P∩C) = 35 (men who have a cellphone and a car)
M(M∩C) = 40 (men who are married and have a car)
M(M∩C∩P) = 30 (men who are married, have a car, and have a cellphone)
Now, let's fill in the Venn diagram step-by-step based on this information:
1. Start by placing M = 150 inside a rectangular shape to represent all men in the town.
2. Draw two overlapping circles within the rectangle to represent sets C (car) and P (cellphone).
3. Place the number 55 (M∩P) in the intersection of sets M and P.
4. Place the number 70 (M(P)) in the remaining part of set P outside the intersection.
5. Place the number 35 (M(P∩C)) in the remaining part of set C outside the intersection.
6. Place the number 75 (M(C)) in the remaining part of set C outside the intersection.
7. Place the number 40 (M(M∩C)) in the intersection of sets M and C.
8. Finally, place the number 30 (M(M∩C∩P)) in the intersection of all three sets.
Now, to find the number of single men who do not have a car, we need to calculate S(∩[C']) (Men who are single and do not have a car):
S(∩[C']) = M - S(∩C) = M - (M(M∩C) + S(∩[C∩M]))
S(∩[C']) = 150 - (40 + 30) = 80.
So, there are 80 single men who do not have a car.
To find the number of single men who do not have either a car or a cellphone, we need to calculate S(∩[C'∩P']) (Men who are single and do not have a car or cellphone):
S(∩[C'∩P']) = M - S(∩C∩P) = M - (M(M∩C∩P) + S(∩[C∩P∩M]))
S(∩[C'∩P']) = 150 - (30 + 40) = 80.
So, there are also 80 single men who do not have either a car or a cellphone.
I hope this helps! If you have any further questions, please let me know.
Sure! Let's break down the information given and set it up using a Venn diagram.
First, let's define the sets:
M = Set of all men in the town
Mᴹ = Set of married men
Mᶜ = Set of single men
C = Set of men who own a car
P = Set of men who have a cell phone
We have the following information:
n(Mᴹ) = 85 (Number of men who are married)
n(P) = 70 (Number of men who have a cell phone)
n(C) = 75 (Number of men who own a car)
n(Mᴹ∩P) = 55 (Number of men who are married and have a cell phone)
n(P∩C) = 35 (Number of men who have a cell phone and a car)
n(Mᴹ∩C) = 40 (Number of men who are married and own a car)
n(Mᴹ∩P∩C) = 30 (Number of men who are married, have a cell phone, and own a car)
Using the information given above, let's populate our Venn diagram step by step:
Step 1: Start with the center of the Venn diagram and fill in the overlapping region of Mᴹ∩P∩C with 30.
Step 2: Fill in the remaining overlapping regions using the information given. Since we already filled in the overlapping region of Mᴹ∩P∩C, we can subtract it from the other overlapping regions to find their respective values.
- n(Mᴹ∩P) = 55. So, the overlapping region of Mᴹ∩P (excluding Mᴹ∩P∩C) is 55 - 30 = 25.
- n(P∩C) = 35. So, the overlapping region of P∩C (excluding Mᴹ∩P∩C) is 35 - 30 = 5.
- n(Mᴹ∩C) = 40. So, the overlapping region of Mᴹ∩C (excluding Mᴹ∩P∩C) is 40 - 30 = 10.
Step 3: Now, fill in the remaining non-overlapping regions using the information given and the values we have already filled.
- n(M) = 150. Since Mᴹ + Mᶜ = n(M), the number of single men is Mᶜ = 150 - 85 = 65.
- n(P) = 70. Since (Mᴹ∩P) + (Mᶜ∩P) + (P∩C) = n(P), the number of single men with a cell phone is (Mᶜ∩P) = 70 - 25 - 30 = 15.
- n(C) = 75. Since (Mᴹ∩C) + (Mᶜ∩C) + (P∩C) = n(C), the number of single men with a car is (Mᶜ∩C) = 75 - 10 - 30 = 35.
Finally, to answer your questions directly:
1. The number of single men who do not have a car = (Mᶜ∩C) = 35.
2. The number of single men who do not have either a car or a cell phone = Mᶜ - (Mᶜ∩P) = 65 - 15 = 50.
I hope this helps!