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What is the nature of relation R, if R is defined as R = {(x, y) : 2x + y = 41, x, y ∈ N}?
- a)reflexive
- b)symmetric
- c)transitive
- d)None of these
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
What is the nature of relation R, if R is defined as R = {(x, y) : 2x ...
Let us check all the conditions:
Reflexivity:
Let x be an arbitrary element of R. For Reflexive, let y + x
i.e. for any x ∈ R
⇒ 2x + x = 41
Let x be an arbitrary element of R. For Reflexive, let y + x
i.e. for any x ∈ R
⇒ 2x + x = 41
Thus, it is NOT reflexive.
Symmetry:
Let (x, y) ∈ R. Then,
2x + y = 41 Which is not equal to 2y + x = 41
i.e. (y , x) ∉ R
So, R is NOT symmetric.
Let (x, y) ∈ R. Then,
2x + y = 41 Which is not equal to 2y + x = 41
i.e. (y , x) ∉ R
So, R is NOT symmetric.
Transitivity:
Let (x, y) and (y, z) ∈ R
⇒ 2x + y = 41 and 2y + z = 41
which is not equal to 2x + z = 41
i.e. (x , z) ∉ R
Thus, R is NOT transitive.
Let (x, y) and (y, z) ∈ R
⇒ 2x + y = 41 and 2y + z = 41
which is not equal to 2x + z = 41
i.e. (x , z) ∉ R
Thus, R is NOT transitive.
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What is the nature of relation R, if R is defined as R = {(x, y) : 2x ...
Understanding Relation R
The relation R is defined as R = {(x, y) : 2x + y = 41, x, y ∈ N}. To analyze the properties of this relation, we will evaluate its reflexivity, symmetry, and transitivity.
Reflexivity
- A relation R is reflexive if for every element a in the set, (a, a) ∈ R.
- In this case, for any natural number x, we need to check if (x, x) satisfies the equation 2x + x = 41.
- This simplifies to 3x = 41, which has no natural number solution.
- Therefore, R is not reflexive.
Symmetry
- A relation R is symmetric if for every (a, b) ∈ R, (b, a) ∈ R also holds true.
- If we take (x, y) such that 2x + y = 41, then to check symmetry, we would need 2y + x = 41 for (y, x).
- Rearranging gives us x = 41 - 2y, which does not guarantee that both pairs will exist.
- Thus, R is not symmetric.
Transitivity
- A relation R is transitive if for any (a, b) and (b, c) in R, (a, c) must also be in R.
- Given the structure of R, finding pairs (x, y) and (y, z) that satisfy both equations is complex and does not guarantee that (x, z) will also satisfy the equation.
- Therefore, R is not transitive.
Conclusion
Since R does not satisfy any of the properties of reflexivity, symmetry, or transitivity, the correct answer is option 'D' - None of these.
The relation R is defined as R = {(x, y) : 2x + y = 41, x, y ∈ N}. To analyze the properties of this relation, we will evaluate its reflexivity, symmetry, and transitivity.
Reflexivity
- A relation R is reflexive if for every element a in the set, (a, a) ∈ R.
- In this case, for any natural number x, we need to check if (x, x) satisfies the equation 2x + x = 41.
- This simplifies to 3x = 41, which has no natural number solution.
- Therefore, R is not reflexive.
Symmetry
- A relation R is symmetric if for every (a, b) ∈ R, (b, a) ∈ R also holds true.
- If we take (x, y) such that 2x + y = 41, then to check symmetry, we would need 2y + x = 41 for (y, x).
- Rearranging gives us x = 41 - 2y, which does not guarantee that both pairs will exist.
- Thus, R is not symmetric.
Transitivity
- A relation R is transitive if for any (a, b) and (b, c) in R, (a, c) must also be in R.
- Given the structure of R, finding pairs (x, y) and (y, z) that satisfy both equations is complex and does not guarantee that (x, z) will also satisfy the equation.
- Therefore, R is not transitive.
Conclusion
Since R does not satisfy any of the properties of reflexivity, symmetry, or transitivity, the correct answer is option 'D' - None of these.
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