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Chapter 1: Problem 39

Let \(\mathrm{R}\) be the relation over the set \(\mathrm{N} \times \mathrm{N}\) and is defined by \((a, b) R(c, d) \Leftrightarrow a+d=b+c\) Then \(R\) is (a) Reflexive only (b) Symmetric only (c) Transitive only (d) An equivalence relation

Short Answer

Expert verified
The relation R is reflexive, symmetric, and transitive, which means it is an equivalence relation. Therefore, the correct answer is \(\textbf{(d) An equivalence relation}\).

Step by step solution

01

Check Reflexivity

To check if the relation is reflexive, we need to check if (a, b) R (a, b) for all elements in N x N. In other words, we need to check if a+b=b+a. According to the commutative property of addition, a+b=b+a for all the numbers in set N. Therefore, the relation R is reflexive.
02

Check Symmetry

To check if the relation is symmetric, we need to test if (a, b) R (c, d) implies (c, d) R (a, b) for all elements in N x N. Assume (a, b) R (c, d), which implies a+d=b+c. To prove symmetry, we need to check if c+b=d+a. If we rewrite the equality a+d=b+c as c+b=a+d, we get the required condition. Hence, the relation R is symmetric.
03

Check Transitivity

To check if the relation is transitive, we need to test if (a, b) R (c, d) and (c, d) R (e, f) implies (a, b) R (e, f) for all elements in N x N. Assume (a, b) R (c, d), which implies a+d=b+c, and (c, d) R (e, f), which implies c+f=d+e. To prove transitivity, we need to check if a+f=b+e. By adding the two equalities, we obtain a+d+c+f=b+c+d+e, which simplifies to a+f=b+e. Hence, the relation R is transitive. Since the relation R has been proven to be Reflexive, Symmetric, and Transitive, we can conclude that it is an equivalence relation. Therefore, the correct answer is (d) An equivalence relation.

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Most popular questions from this chapter

The relation \(\leq\) on numbers has the following properties. (a) \(a \leq a \forall a \in R\) (Reflexivity) (b) If \(\mathrm{a} \leq \mathrm{b}\) and \(\mathrm{b} \leq \mathrm{a}\) then \(\mathrm{a}=\mathrm{b} \forall \mathrm{a}, \mathrm{b} \in \mathrm{R}\) (Antisymmetry) (c) If \(\mathrm{a} \leq \mathrm{b}\) and \(\mathrm{b} \leq \mathrm{c}\) then \(\mathrm{a} \leq \mathrm{c} \forall \mathrm{a}, \mathrm{b} \in \mathrm{R}\) (Transitivity) Which of the above properties the relation \(\subset\) on \(\mathrm{P}(\mathrm{A})\) has? (a) (i) and (ii) (b) (i) and (iii) (c) (ii) and (iii) (d) (i), (ii) and (iii)

Which one of the following relations on \(R\) is an equivalence relation? (a) a \(\mathrm{R}_{1} \mathrm{~b} \Leftrightarrow|\mathrm{a}|=|\mathrm{b}|\) (b) a \(\mathrm{R}_{2} \mathrm{~b} \Leftrightarrow \mathrm{a} \geq \mathrm{b}\) (c) a \(\mathrm{R}_{3} \mathrm{~b} \Leftrightarrow\) a divides \(\mathrm{b}\) (d) a \(\mathrm{R}_{4} \mathrm{~b} \Leftrightarrow \mathrm{a}<\mathrm{b}\)

A Relation \(R\) is defined in the set of integers I as follows \((\mathrm{x}, \mathrm{y}) \in \mathrm{R}\) if \(\mathrm{x}^{2}+\mathrm{y}^{2}=9\) which of the following is false ? (a) \(\mathrm{R}=\\{(0,3),(0,-3),(3,0),(-3,0)\\}\) (b) Domain of \(\mathrm{R}=\\{-3,0,3\\}\) (c) Range of \(\mathrm{R}=\\{-3,0,3\\}\) (d) None of the above

A function \(\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}\) satisfies the equation \(\mathrm{f}(\mathrm{x}) \mathrm{f}(\mathrm{y})-\mathrm{f}(\mathrm{xy})=\mathrm{x}+\mathrm{y}\) for all \(\mathrm{x}, \mathrm{y} \in \mathrm{R}\) and \(\mathrm{f}(1)>0\), then (a) \(\mathrm{f}(\mathrm{x})=\mathrm{x}+(1 / 2)\) (b) \(\mathrm{f}(\mathrm{x})=(1 / 2) \mathrm{x}+1\) (c) \(\mathrm{f}(\mathrm{x})=\mathrm{x}+1\) (d) \(\mathrm{f}(\mathrm{x})=(1 / 2) \mathrm{x}-1\)

If A is the set of even natural numbers less than 8 and \(\mathrm{B}\) is the set of prime numbers less than 7, then the number of relations from \(\mathrm{A}\) to \(\mathrm{B}\) is (a) \(2^{9}\) (b) \(9^{2}\) (c) \(3^{2}\) (d) \(2^{9}-1\)
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