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130 जना विद्यार्थीहरूसँग ल्यापटप भएको, 150 जना विद्यार्थीहरूसँग ट्याब्लेट भएको, र 50 जना विद्यार्थीहरूसँग दुवै उपकरण भएको पत्ता लाग्यो।

(a) जम्मा विद्यार्थीहरूको समूह [tex]$U$[/tex] भए, [tex]$U$[/tex] को गणनात्मकता लेख्नुहोस्।
If the set of total students is [tex][tex]$U$[/tex][/tex], write the cardinality of [tex]$U$[/tex].

(b) माथिको जानकारीलाई वेन-आरेखमा प्रस्तुत गर्नुहोस्।
Present the above information in a Venn diagram.

(c) दुवै उपकरण नहुने विद्यार्थीहरूको संख्या पत्ता लगाउनुहोस्।
Find the number of students who do not have either equipment.

(d) कुनै एउटा मात्र उपकरण हुने र उपकरण नै नहुने विद्यार्थीहरूको संख्यामा कति फरक छ, तुलना गर्नुहोस्।
What is the difference between the number of students who have only one equipment and those who do not have any equipment?

Answer:
(a) 250
(c) 20



Answer :

GinnyAnswer
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.

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