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Sagot :
Certainly! Let's tackle each part of the question step by step:
### Part a ###
Are the given sets equal or equivalent? Give reason:
- Sets:
- [tex]\( M = \{ \text{Rajkumar}, \text{Uttam}, \text{Deependra}, \text{Bijay}, \text{Nagendra}\} \)[/tex]
- [tex]\( F = \{ \text{Tripti}, \text{Pabina}, \text{Sunita}, \text{Sushma}, \text{Rita} \} \)[/tex]
- Equality: Two sets are considered equal if they contain exactly the same elements. Here, it is clear that:
- The elements of [tex]\( M \)[/tex] are different from the elements of [tex]\( F \)[/tex].
- Therefore, [tex]\( M \)[/tex] and [tex]\( F \)[/tex] are not equal.
- Equivalence: Two sets are considered equivalent if they have the same number of elements. Here:
- Set [tex]\( M \)[/tex] has 5 elements.
- Set [tex]\( F \)[/tex] also has 5 elements.
- Since both sets have the same number of elements, [tex]\( M \)[/tex] and [tex]\( F \)[/tex] are equivalent.
So, the sets [tex]\( M \)[/tex] and [tex]\( F \)[/tex] are not equal but they are equivalent.
Answer: The given sets are not equal, but they are equivalent because they have the same number of elements.
### Part b ###
How many subsets are possible from the set ' [tex]\( M \)[/tex] '? Calculate using formula.
- Set: [tex]\( M = \{\text{Rajkumar}, \text{Uttam}, \text{Deependra}, \text{Bijay}, \text{Nagendra} \} \)[/tex]
- Number of subsets: The number of subsets of a set with [tex]\( n \)[/tex] elements is given by the formula [tex]\( 2^n \)[/tex].
- In this case, [tex]\( M \)[/tex] has 5 elements.
So, the number of subsets of [tex]\( M \)[/tex] is [tex]\( 2^5 = 32 \)[/tex].
Answer: The number of subsets possible from the set [tex]\( M \)[/tex] is 32.
### Part c ###
Is [tex]\( A = \{\text{Bina}, \text{Meenu}, \text{Karishma}\} \)[/tex] a proper subset or improper subset of [tex]\( B = \{\text{Tripti}, \text{Pabina}, \text{Sunita}, \text{Sushma}, \text{Rita} \} \)[/tex]? Why?
- Subset Definitions:
- A proper subset is a subset that is strictly contained within another set and is not equal to that set.
- An improper subset is a subset that can be equal to the set it is being compared to.
- Sets:
- [tex]\( A = \{\text{Bina}, \text{Meenu}, \text{Karishma}\} \)[/tex]
- [tex]\( B = \{\text{Tripti}, \text{Pabina}, \text{Sunita}, \text{Sushma}, \text{Rita}\} \)[/tex]
- Check Proper Subset:
- For [tex]\( A \)[/tex] to be a proper subset of [tex]\( B \)[/tex], every element of [tex]\( A \)[/tex] must be in [tex]\( B \)[/tex] and [tex]\( A \)[/tex] must not equal [tex]\( B \)[/tex].
- Here, none of the elements of [tex]\( A \)[/tex] (\text{Bina}, \text{Meenu}, \text{Karishma}) are present in [tex]\( B \)[/tex].
- Therefore, [tex]\( A \)[/tex] is not a proper subset of [tex]\( B \)[/tex] because [tex]\( A \)[/tex] does not contain any elements from [tex]\( B \)[/tex].
- Check Improper Subset:
- [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are clearly different.
- An improper subset would imply that [tex]\( A \)[/tex] is exactly equal to [tex]\( B \)[/tex].
So, [tex]\( A \)[/tex] is not a proper subset of [tex]\( B \)[/tex] and not an improper subset either, as [tex]\( A \)[/tex] is not contained within [tex]\( B \)[/tex] at all and also not equal to [tex]\( B \)[/tex].
Answer: [tex]\( A \)[/tex] is neither a proper subset nor an improper subset of [tex]\( B \)[/tex] because none of the elements of [tex]\( A \)[/tex] are present in [tex]\( B \)[/tex].
### Part a ###
Are the given sets equal or equivalent? Give reason:
- Sets:
- [tex]\( M = \{ \text{Rajkumar}, \text{Uttam}, \text{Deependra}, \text{Bijay}, \text{Nagendra}\} \)[/tex]
- [tex]\( F = \{ \text{Tripti}, \text{Pabina}, \text{Sunita}, \text{Sushma}, \text{Rita} \} \)[/tex]
- Equality: Two sets are considered equal if they contain exactly the same elements. Here, it is clear that:
- The elements of [tex]\( M \)[/tex] are different from the elements of [tex]\( F \)[/tex].
- Therefore, [tex]\( M \)[/tex] and [tex]\( F \)[/tex] are not equal.
- Equivalence: Two sets are considered equivalent if they have the same number of elements. Here:
- Set [tex]\( M \)[/tex] has 5 elements.
- Set [tex]\( F \)[/tex] also has 5 elements.
- Since both sets have the same number of elements, [tex]\( M \)[/tex] and [tex]\( F \)[/tex] are equivalent.
So, the sets [tex]\( M \)[/tex] and [tex]\( F \)[/tex] are not equal but they are equivalent.
Answer: The given sets are not equal, but they are equivalent because they have the same number of elements.
### Part b ###
How many subsets are possible from the set ' [tex]\( M \)[/tex] '? Calculate using formula.
- Set: [tex]\( M = \{\text{Rajkumar}, \text{Uttam}, \text{Deependra}, \text{Bijay}, \text{Nagendra} \} \)[/tex]
- Number of subsets: The number of subsets of a set with [tex]\( n \)[/tex] elements is given by the formula [tex]\( 2^n \)[/tex].
- In this case, [tex]\( M \)[/tex] has 5 elements.
So, the number of subsets of [tex]\( M \)[/tex] is [tex]\( 2^5 = 32 \)[/tex].
Answer: The number of subsets possible from the set [tex]\( M \)[/tex] is 32.
### Part c ###
Is [tex]\( A = \{\text{Bina}, \text{Meenu}, \text{Karishma}\} \)[/tex] a proper subset or improper subset of [tex]\( B = \{\text{Tripti}, \text{Pabina}, \text{Sunita}, \text{Sushma}, \text{Rita} \} \)[/tex]? Why?
- Subset Definitions:
- A proper subset is a subset that is strictly contained within another set and is not equal to that set.
- An improper subset is a subset that can be equal to the set it is being compared to.
- Sets:
- [tex]\( A = \{\text{Bina}, \text{Meenu}, \text{Karishma}\} \)[/tex]
- [tex]\( B = \{\text{Tripti}, \text{Pabina}, \text{Sunita}, \text{Sushma}, \text{Rita}\} \)[/tex]
- Check Proper Subset:
- For [tex]\( A \)[/tex] to be a proper subset of [tex]\( B \)[/tex], every element of [tex]\( A \)[/tex] must be in [tex]\( B \)[/tex] and [tex]\( A \)[/tex] must not equal [tex]\( B \)[/tex].
- Here, none of the elements of [tex]\( A \)[/tex] (\text{Bina}, \text{Meenu}, \text{Karishma}) are present in [tex]\( B \)[/tex].
- Therefore, [tex]\( A \)[/tex] is not a proper subset of [tex]\( B \)[/tex] because [tex]\( A \)[/tex] does not contain any elements from [tex]\( B \)[/tex].
- Check Improper Subset:
- [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are clearly different.
- An improper subset would imply that [tex]\( A \)[/tex] is exactly equal to [tex]\( B \)[/tex].
So, [tex]\( A \)[/tex] is not a proper subset of [tex]\( B \)[/tex] and not an improper subset either, as [tex]\( A \)[/tex] is not contained within [tex]\( B \)[/tex] at all and also not equal to [tex]\( B \)[/tex].
Answer: [tex]\( A \)[/tex] is neither a proper subset nor an improper subset of [tex]\( B \)[/tex] because none of the elements of [tex]\( A \)[/tex] are present in [tex]\( B \)[/tex].
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