JEE Exam > JEE Questions > Let A = {1,2,3,4,5,6,7}. P = {1,2}, Q = {3, 7...
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Let A = {1,2,3,4,5,6,7}. P = {1,2}, Q = {3, 7}. Write the elements of the set R so that P, Q and R form a partition that results in equivalence relation.
- a){4,5,6}
- b){0}
- c){1,2,3,4,5,6,7}
- d){ }
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
Let A = {1,2,3,4,5,6,7}. P = {1,2}, Q = {3, 7}. Write the elements of ...
Solution:
Equivalence relation is a relation that is reflexive, symmetric and transitive. A partition of a set is a collection of non-empty subsets whose union is the original set and whose members are pairwise disjoint. Let's find the set R so that P, Q and R form a partition that results in equivalence relation.
Reflexive property: Every element is related to itself.
Symmetric property: If a is related to b, then b is related to a.
Transitive property: If a is related to b and b is related to c, then a is related to c.
Partition: P, Q and R are pairwise disjoint. Their union is A.
Elements of the set R: {4, 5, 6}
Explanation:
We can choose the set R as {4, 5, 6} because it is disjoint from both P and Q. This means that the sets P, Q and R are pairwise disjoint. Also, their union is A, which means that every element of A belongs to exactly one of the sets P, Q and R.
Now, let's check whether the partition formed by P, Q and R results in an equivalence relation.
Reflexive property: Every element is related to itself.
- For any element x in A, x belongs to exactly one of the sets P, Q and R.
- If x belongs to P or Q, then x is related to itself because P and Q are both reflexive.
- If x belongs to R, then x is related to itself because R is a set of elements.
Symmetric property: If a is related to b, then b is related to a.
- For any elements a and b in A, a belongs to exactly one of the sets P, Q and R, and b belongs to exactly one of the sets P, Q and R.
- If a and b belong to the same set (P, Q or R), then the relation between them is symmetric because P, Q and R are all symmetric.
- If a and b belong to different sets, then they are not related. Therefore, the relation between them is trivially symmetric.
Transitive property: If a is related to b and b is related to c, then a is related to c.
- For any elements a, b and c in A, a belongs to exactly one of the sets P, Q and R, b belongs to exactly one of the sets P, Q and R, and c belongs to exactly one of the sets P, Q and R.
- If a and b belong to the same set and b and c belong to the same set, then a and c belong to the same set by the transitive property of P, Q and R.
- If a and b belong to different sets or b and c belong to different sets, then a and c are not related. Therefore, the relation between them is trivially transitive.
Therefore, the partition formed by P, Q and R results in an equivalence relation.
Equivalence relation is a relation that is reflexive, symmetric and transitive. A partition of a set is a collection of non-empty subsets whose union is the original set and whose members are pairwise disjoint. Let's find the set R so that P, Q and R form a partition that results in equivalence relation.
Reflexive property: Every element is related to itself.
Symmetric property: If a is related to b, then b is related to a.
Transitive property: If a is related to b and b is related to c, then a is related to c.
Partition: P, Q and R are pairwise disjoint. Their union is A.
Elements of the set R: {4, 5, 6}
Explanation:
We can choose the set R as {4, 5, 6} because it is disjoint from both P and Q. This means that the sets P, Q and R are pairwise disjoint. Also, their union is A, which means that every element of A belongs to exactly one of the sets P, Q and R.
Now, let's check whether the partition formed by P, Q and R results in an equivalence relation.
Reflexive property: Every element is related to itself.
- For any element x in A, x belongs to exactly one of the sets P, Q and R.
- If x belongs to P or Q, then x is related to itself because P and Q are both reflexive.
- If x belongs to R, then x is related to itself because R is a set of elements.
Symmetric property: If a is related to b, then b is related to a.
- For any elements a and b in A, a belongs to exactly one of the sets P, Q and R, and b belongs to exactly one of the sets P, Q and R.
- If a and b belong to the same set (P, Q or R), then the relation between them is symmetric because P, Q and R are all symmetric.
- If a and b belong to different sets, then they are not related. Therefore, the relation between them is trivially symmetric.
Transitive property: If a is related to b and b is related to c, then a is related to c.
- For any elements a, b and c in A, a belongs to exactly one of the sets P, Q and R, b belongs to exactly one of the sets P, Q and R, and c belongs to exactly one of the sets P, Q and R.
- If a and b belong to the same set and b and c belong to the same set, then a and c belong to the same set by the transitive property of P, Q and R.
- If a and b belong to different sets or b and c belong to different sets, then a and c are not related. Therefore, the relation between them is trivially transitive.
Therefore, the partition formed by P, Q and R results in an equivalence relation.
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Community Answer
Let A = {1,2,3,4,5,6,7}. P = {1,2}, Q = {3, 7}. Write the elements of ...
P{1,2} union Q{3,7} union R{}=A{1,2,3,4,5,6,7}
therefore R={4,5,6}
therefore R={4,5,6}
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Let A = {1,2,3,4,5,6,7}. P = {1,2}, Q = {3, 7}. Write the elements of the set R so that P, Q and R form a partition that results in equivalence relation.a){4,5,6}b){0}c){1,2,3,4,5,6,7}d){ }Correct answer is option 'A'. Can you explain this answer?
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Let A = {1,2,3,4,5,6,7}. P = {1,2}, Q = {3, 7}. Write the elements of the set R so that P, Q and R form a partition that results in equivalence relation.a){4,5,6}b){0}c){1,2,3,4,5,6,7}d){ }Correct answer is option 'A'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let A = {1,2,3,4,5,6,7}. P = {1,2}, Q = {3, 7}. Write the elements of the set R so that P, Q and R form a partition that results in equivalence relation.a){4,5,6}b){0}c){1,2,3,4,5,6,7}d){ }Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let A = {1,2,3,4,5,6,7}. P = {1,2}, Q = {3, 7}. Write the elements of the set R so that P, Q and R form a partition that results in equivalence relation.a){4,5,6}b){0}c){1,2,3,4,5,6,7}d){ }Correct answer is option 'A'. Can you explain this answer?.
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