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Equivalence relation Video Lecture | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET
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FAQs on Equivalence relation Video Lecture - Mathematics for IIT JAM, GATE, CSIR NET, UGC NET
| 1. What is an equivalence relation? |  |
Ans. An equivalence relation is a relation that is reflexive, symmetric, and transitive. In other words, it is a relation that satisfies three properties: every element is related to itself (reflexive property), if element A is related to element B, then element B is also related to element A (symmetric property), and if element A is related to element B and element B is related to element C, then element A is also related to element C (transitive property).
| 2. How do you determine if a relation is an equivalence relation? | |
Ans. To determine if a relation is an equivalence relation, you need to check if it satisfies three properties: reflexivity, symmetry, and transitivity. If for every element in the set, it is related to itself (reflexivity), for every pair of related elements, the reverse relation also holds (symmetry), and for any three related elements, the first element is also related to the third element (transitivity), then the relation is an equivalence relation.
| 3. Can you provide an example of an equivalence relation? | |
Ans. Yes, one example of an equivalence relation is the relation of equality on the set of integers. In this relation, any integer is related to itself (reflexivity), if two integers are equal, then they are related to each other (symmetry), and if two integers are equal and a third integer is also equal to the second integer, then the first integer is also equal to the third integer (transitivity).
| 4. What are the applications of equivalence relations in mathematics? | |
Ans. Equivalence relations have various applications in mathematics. They are used in algebraic structures, such as groups and rings, to define equivalence classes and quotient structures. Equivalence relations also play a crucial role in set theory, where they are used to partition sets into disjoint subsets. Additionally, equivalence relations are utilized in graph theory to study connectedness and in topology to define homeomorphism between spaces.
| 5. How are equivalence relations related to partitions? | |
Ans. Equivalence relations and partitions are closely related concepts. An equivalence relation on a set divides the set into disjoint subsets called equivalence classes. These equivalence classes form a partition of the set, where each element belongs to exactly one equivalence class. Conversely, any partition of a set defines an equivalence relation, where two elements are related if and only if they belong to the same subset in the partition. Thus, equivalence relations and partitions provide alternative ways to describe the same underlying structure.
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Dec 22, 2024 Last updated |
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