Maths · Relations, type of relations, equivalence relations
Let be a relation defined on the set of natural numbers, as \in \boldsymbol{N} \
Let \( \rho \) be a relation defined on \( N, \) the set of natural numbers, as \( \boldsymbol{\rho}=\{(\boldsymbol{x}, \boldsymbol{y}) \in \boldsymbol{N} \times \boldsymbol{N}: \mathbf{2} \boldsymbol{x}+\boldsymbol{y}=\mathbf{4 1}\} \) then
- A. \( \rho \) is an equivalence relation
- B. \( \rho \) is only reflexive relation
- C. \( \rho \) is only symmetric relation
- D. \( \rho \) is not transitive
Step-by-step solution
The relation ρ is defined by 2x+y=41. For reflexivity, we need 3x=41, which has no natural solution, so not reflexive. For symmetry, if (x,y)∈ρ then 2x+y=41 but 2y+x=41 implies x=y, and no such natural x exists, so not symmetric. For transitivity, take (11,19) and (19,3) in ρ, then (11,3) gives 2*11+3=25≠41, so transitivity fails. Therefore ρ is not transitive, and option D is correct.
Related MCQs
- Which of the following is NOT equivalent to…
- Which of the following statements is the contrapositive of the statement, You win the game if you know the rules but are not overconfident.…
- The contrapositive statement of statement "If is prime number, then is odd" is…
- Assertion The relation given by ,(\mathbf{4}, \mathbf{2}),(\mathbf{2}, \mathbf{4}),(\mathbf{2}, \mathbf{3}),(\mathbf{3}, \mathbf{1})\} \) on…
- Which of the following statements is the inverse of "Our pond floods whenever there is a thunderstorm."?…