Students are trying to fix thumb pins on a graph sheet on the class notice board. A relation \(R\) is defined on the set \(A\) of thumb pins as
\[ R = \{(P, Q) : P, Q \in A \text{ and distance between } P \text{ and } Q \text{ is less than } 3 \text{ cm}\} \]
Determine if \(R\) is an equivalence relation.
Students are trying to fix thumb pins on a graph sheet on the class notice board. A relation \(R\) is defined on the set \(A\) of thumb pins as
\[ R = \{(P, Q) : P, Q \in A \text{ and distance between } P \text{ and } Q \text{ is less than } 3 \text{ cm}\} \]
Determine if \(R\) is an equivalence relation.
Show Hint
Solution and Explanation
A relation is an equivalence relation if it is reflexive, symmetric and transitive.
We check all three for \( R \).
Step 2: Reflexive:
For any pin \( P \), the distance between \( P \) and itself is \( 0 \) cm. This is less than \( 3 \) cm.
So \( (P, P) \in R \) and \( R \) is reflexive.
Step 3: Symmetric:
The distance from \( P \) to \( Q \) equals the distance from \( Q \) to \( P \).
So if \( (P, Q) \in R \), then \( (Q, P) \in R \). Hence \( R \) is symmetric.
Step 4: Transitive fails:
Take three pins \( P, Q, S \) on one line. Put \( Q \) at \( 2 \) cm from \( P \) and \( S \) at \( 2 \) cm from \( Q \), on the far side.
Then \( (P, Q) \in R \) and \( (Q, S) \in R \). But the distance from \( P \) to \( S \) is \( 4 \) cm, which is not less than \( 3 \) cm.
So \( (P, S) \notin R \) and \( R \) is not transitive.
Final Answer:
\( R \) is reflexive and symmetric but not transitive. \[ \boxed{R \text{ is not an equivalence relation}} \]
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Show that the relation R in the set R of real numbers, defined as
R = {(a, b): a ≤ b2 } is neither reflexive nor symmetric nor transitive.Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as
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Assertion (A) : Let \(R\) be a relation on set \(A\) of all books in a school library given by \(R = \{(x, y) : x, y \in A,\ x \text{ and } y \text{ have the same number of pages}\}\) and \(S\) is a relation on set \(A\) from the same library such that \(S = \{(p, q) : p \text{ and } q \text{ are Mathematics books}\}\). Also, \(R\cap S \neq \phi\). Then, \(R\cap S\) is an equivalence relation.
Reason (R) : The intersection of two equivalence relations on a set is an equivalence relation on the set.
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