At Westonci.ca, we make it easy for you to get the answers you need from a community of knowledgeable individuals. Discover comprehensive solutions to your questions from a wide network of experts on our user-friendly platform. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
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].
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.
