Discrete Mathematics previous year-GATE 2025 Equivalence relation Let R and S be any two equivalence
Discrete Mathematics previous year-GATE 2025 Equivalence relation Let R and S be any two equivalence
In the context of GATE (Graduate Aptitude Test in Engineering) examinations, understanding the properties of equivalence relations is crucial. An equivalence relation on a set AA is a relation that is reflexive, symmetric, and transitive.
Key Properties of Equivalence Relations:
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Reflexivity: Every element is related to itself. For all a∈Aa \in A, (a,a)∈R(a, a) \in R.
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Symmetry: If an element aa is related to an element bb, then bb is also related to aa. For all a,b∈Aa, b \in A, if (a,b)∈R(a, b) \in R, then (b,a)∈R(b, a) \in R.
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Transitivity: If an element aa is related to bb, and bb is related to cc, then aa is related to cc. For all a,b,c∈Aa, b, c \in A, if (a,b)∈R(a, b) \in R and (b,c)∈R(b, c) \in R, then (a,c)∈R(a, c) \in R.
Intersection of Equivalence Relations:
When considering two equivalence relations RR and SS on a set AA, their intersection R∩SR \cap S is also an equivalence relation. This is because the intersection of two reflexive, symmetric, and transitive relations retains these properties.
Union of Equivalence Relations:
However, the union R∪SR \cup S of two equivalence relations is not necessarily an equivalence relation. While the union of two reflexive and symmetric relations remains reflexive and symmetric, it may fail to be transitive.
Example Illustrating Non-Transitivity:
Consider a set A={1,2,3}A = \{1, 2, 3\} with two equivalence relations RR and SS:
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R={(1,2),(2,1)}R = \{(1, 2), (2, 1)\}
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S={(2,3),(3,2)}S = \{(2, 3), (3, 2)\}
Their union is:
R∪S={(1,2),(2,1),(2,3),(3,2)}R \cup S = \{(1, 2), (2, 1), (2, 3), (3, 2)\}
In this union, we have (1,2)∈R∪S(1, 2) \in R \cup S and (2,3)∈R∪S(2, 3) \in R \cup S, but (1,3)∉R∪S(1, 3) \notin R \cup S. This lack of transitivity means R∪SR \cup S is not an equivalence relation.
GATE Examination Insight:
A relevant GATE question from 2005 asked:
*”Let RR and SS be any two equivalence relations on a non-empty set AA. Which one of the following statements is TRUE?
A. R∪SR \cup S and R∩SR \cap S are both equivalence relations.
B. R∪SR \cup S is an equivalence relation.
C. R∩SR \cap S is an equivalence relation.
D. Neither R∪SR \cup S nor R∩SR \cap S is an equivalence relation.”*
The correct answer is C: R∩SR \cap S is an equivalence relation.
Conclusion:
Understanding the behavior of unions and intersections of equivalence relations is essential for discrete mathematics and examinations like GATE. While intersections of equivalence relations remain equivalence relations, unions do not necessarily preserve the transitivity property required for equivalence relations.
Contents [hide]
- 1 Discrete Mathematics previous year-GATE 2025 Equivalence relation Let R and S be any two equivalence
- 2 Discrete Mathematics for Computer Science
- 3
Key Concepts: - 4
GATE Previous Year Question Example: - 5 Discrete Mathematics previous year-GATE 2025 Equivalence relation Let R and S be any two equivalence
- 6 Data Science and Artificial Intelligence (DA) – GATE 2025
Discrete Mathematics previous year-GATE 2025 Equivalence relation Let R and S be any two equivalence
Discrete Mathematics for Computer Science
In Discrete Mathematics, particularly in the context of GATE (Graduate Aptitude Test in Engineering) examinations, understanding the properties of equivalence relations and their combinations is crucial.
Key Concepts:
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Equivalence Relation: A relation RR on a set AA is called an equivalence relation if it is reflexive, symmetric, and transitive.
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Intersection of Equivalence Relations (R∩SR \cap S): If RR and SS are equivalence relations on the same set AA, then their intersection R∩SR \cap S is also an equivalence relation. This is because the intersection of reflexive, symmetric, and transitive relations retains these properties.
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Union of Equivalence Relations (R∪SR \cup S): The union R∪SR \cup S of two equivalence relations RR and SS on the same set AA may not be an equivalence relation. While the union of reflexive and symmetric relations remains reflexive and symmetric, it may fail to be transitive.
GATE Previous Year Question Example:
GATE CSE 2005, Question 42:
Let RR and SS be any two equivalence relations on a non-empty set AA. Which one of the following statements is TRUE?
Options:
A) R∪SR \cup S is always an equivalence relation.
B) R∩SR \cap S is always an equivalence relation.
C) R∘SR \circ S is always an equivalence relation.
D) R−SR – S is always an equivalence relation.
Correct Answer: B) R∩SR \cap S is always an equivalence relation.
Explanation:
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Option A: As mentioned, R∪SR \cup S may not be transitive, hence not always an equivalence relation.
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Option B: The intersection of two equivalence relations retains reflexivity, symmetry, and transitivity, thus always an equivalence relation.
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Option C: The composition R∘SR \circ S may not be symmetric or transitive, hence not necessarily an equivalence relation.
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Option D: The difference R−SR – S may not retain any of the three required properties, so it’s not guaranteed to be an equivalence relation.
For a more in-depth understanding and additional examples, you might find the following video helpful:
Feel free to ask if you need further clarification or assistance with other topics!