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Sagot :
Let's go through each part of the problem systematically with detailed explanations.
### (a) Cardinality of the Set U
The total number of students in the class is given as 250. In set notation, this total set of students is denoted by [tex]\( U \)[/tex].
So, the cardinality of the set [tex]\( U \)[/tex] is:
[tex]\[ |U| = 250 \][/tex]
### (b) Present the Above Information in a Venn Diagram
To present the information in a Venn diagram:
- Draw a rectangle to represent the universal set [tex]\( U \)[/tex] which contains all 250 students.
- Inside the rectangle, draw two overlapping circles:
- Label one circle "Laptop" and the other "Tablet".
- The overlapping area represents students who have both a laptop and a tablet.
From the given information:
- Total students with a laptop = 130
- Total students with a tablet = 150
- Students with both a laptop and a tablet = 50
Place these values in the corresponding areas of the Venn diagram.
### (c) Number of Students with Neither Equipment
To find the number of students who have neither a laptop nor a tablet, use the principle of inclusion and exclusion:
[tex]\[ \text{Students with neither} = \text{Total students} - (\text{Students with laptop} + \text{Students with tablet} - \text{Students with both}) \][/tex]
Substituting the given values:
[tex]\[ \text{Students with neither} = 250 - (130 + 150 - 50) \][/tex]
Calculate the expression inside the parentheses first:
[tex]\[ 130 + 150 - 50 = 230 \][/tex]
So:
[tex]\[ \text{Students with neither} = 250 - 230 = 20 \][/tex]
Therefore, the number of students who do not have either piece of equipment is:
[tex]\[ 20 \][/tex]
### (d) Difference between Students with Only One Equipment and Those with Neither
First, find the number of students with only one type of equipment:
- Students with only a laptop:
[tex]\[ \text{Students with only a laptop} = \text{Students with laptop} - \text{Students with both} \][/tex]
[tex]\[ \text{Students with only a laptop} = 130 - 50 = 80 \][/tex]
- Students with only a tablet:
[tex]\[ \text{Students with only a tablet} = \text{Students with tablet} - \text{Students with both} \][/tex]
[tex]\[ \text{Students with only a tablet} = 150 - 50 = 100 \][/tex]
Add these to find the total number of students with only one piece of equipment:
[tex]\[ \text{Students with only one equipment} = \text{Students with only a laptop} + \text{Students with only a tablet} \][/tex]
[tex]\[ \text{Students with only one equipment} = 80 + 100 = 180 \][/tex]
Now, compare this with the number of students who have neither piece of equipment:
[tex]\[ \text{Difference} = \text{Students with only one equipment} - \text{Students with neither} \][/tex]
[tex]\[ \text{Difference} = 180 - 20 = 160 \][/tex]
Therefore, the difference between the number of students who have only one type of equipment and those who do not have any equipment is:
[tex]\[ 160 \][/tex]
### Summary
- (a) The cardinality of set [tex]\( U \)[/tex] is 250.
- (c) The number of students who do not have both the equipment is 20.
- (d) The difference between the number of students who have only one type of equipment and those who do not have any equipment is 160.
### (a) Cardinality of the Set U
The total number of students in the class is given as 250. In set notation, this total set of students is denoted by [tex]\( U \)[/tex].
So, the cardinality of the set [tex]\( U \)[/tex] is:
[tex]\[ |U| = 250 \][/tex]
### (b) Present the Above Information in a Venn Diagram
To present the information in a Venn diagram:
- Draw a rectangle to represent the universal set [tex]\( U \)[/tex] which contains all 250 students.
- Inside the rectangle, draw two overlapping circles:
- Label one circle "Laptop" and the other "Tablet".
- The overlapping area represents students who have both a laptop and a tablet.
From the given information:
- Total students with a laptop = 130
- Total students with a tablet = 150
- Students with both a laptop and a tablet = 50
Place these values in the corresponding areas of the Venn diagram.
### (c) Number of Students with Neither Equipment
To find the number of students who have neither a laptop nor a tablet, use the principle of inclusion and exclusion:
[tex]\[ \text{Students with neither} = \text{Total students} - (\text{Students with laptop} + \text{Students with tablet} - \text{Students with both}) \][/tex]
Substituting the given values:
[tex]\[ \text{Students with neither} = 250 - (130 + 150 - 50) \][/tex]
Calculate the expression inside the parentheses first:
[tex]\[ 130 + 150 - 50 = 230 \][/tex]
So:
[tex]\[ \text{Students with neither} = 250 - 230 = 20 \][/tex]
Therefore, the number of students who do not have either piece of equipment is:
[tex]\[ 20 \][/tex]
### (d) Difference between Students with Only One Equipment and Those with Neither
First, find the number of students with only one type of equipment:
- Students with only a laptop:
[tex]\[ \text{Students with only a laptop} = \text{Students with laptop} - \text{Students with both} \][/tex]
[tex]\[ \text{Students with only a laptop} = 130 - 50 = 80 \][/tex]
- Students with only a tablet:
[tex]\[ \text{Students with only a tablet} = \text{Students with tablet} - \text{Students with both} \][/tex]
[tex]\[ \text{Students with only a tablet} = 150 - 50 = 100 \][/tex]
Add these to find the total number of students with only one piece of equipment:
[tex]\[ \text{Students with only one equipment} = \text{Students with only a laptop} + \text{Students with only a tablet} \][/tex]
[tex]\[ \text{Students with only one equipment} = 80 + 100 = 180 \][/tex]
Now, compare this with the number of students who have neither piece of equipment:
[tex]\[ \text{Difference} = \text{Students with only one equipment} - \text{Students with neither} \][/tex]
[tex]\[ \text{Difference} = 180 - 20 = 160 \][/tex]
Therefore, the difference between the number of students who have only one type of equipment and those who do not have any equipment is:
[tex]\[ 160 \][/tex]
### Summary
- (a) The cardinality of set [tex]\( U \)[/tex] is 250.
- (c) The number of students who do not have both the equipment is 20.
- (d) The difference between the number of students who have only one type of equipment and those who do not have any equipment is 160.
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