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For real numbers x and y, we write xRy <=> x - y + √2 is an irrational number. Then, the relation R is
- a)reflexive
- b)symmetric
- c)transitive
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
For real numbers x and y, we write xRy <=> x - y + √2 is a...
For any x ∈ R, we have x - x +√2 = √2 an irrational number.
implies xRx for all x. So, R is reflexive.
R is not symmetric, because
and 
R is not transitive also because 
and 
but 
.
implies xRx for all x. So, R is reflexive.
R is not symmetric, because
and R is not transitive also because and but .Most Upvoted Answer
For real numbers x and y, we write xRy <=> x - y + √2 is a...
For any x ∈ R, we have x - x +√2 = √2 an irrational number.
implies xRx for all x. So, R is reflexive.
R is not symmetric, because
and 
R is not transitive also because 
and 
but 
.
implies xRx for all x. So, R is reflexive.
R is not symmetric, because
and R is not transitive also because and but .Free Test
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Community Answer
For real numbers x and y, we write xRy <=> x - y + √2 is a...
Reflexive Property:
- For a relation to be reflexive, every element must be related to itself.
- In this case, let's consider x - x + √2, which simplifies to √2.
- Since √2 is an irrational number, xRx holds true for all real numbers x.
Symmetric Property:
- For a relation to be symmetric, if x is related to y, then y must also be related to x.
- Let's consider xRy, which means x - y + √2 is irrational.
- Now, yRx would imply y - x + √2 is irrational.
- However, y - x + √2 simplifies to -(x - y + √2), which is the negation of x - y + √2.
- Since the negation of an irrational number is not necessarily irrational, the relation is not symmetric.
Transitive Property:
- For a relation to be transitive, if x is related to y and y is related to z, then x must be related to z.
- Consider xRy and yRz, which mean x - y + √2 and y - z + √2 are irrational.
- Now, (x - z) + 2√2 = (x - y + √2) + (y - z + √2) is irrational.
- Since 2√2 is irrational and the sum of two irrational numbers is not guaranteed to be irrational, the relation is not transitive.
Therefore, the relation R is reflexive as every real number is related to itself, but it is not symmetric or transitive.
- For a relation to be reflexive, every element must be related to itself.
- In this case, let's consider x - x + √2, which simplifies to √2.
- Since √2 is an irrational number, xRx holds true for all real numbers x.
Symmetric Property:
- For a relation to be symmetric, if x is related to y, then y must also be related to x.
- Let's consider xRy, which means x - y + √2 is irrational.
- Now, yRx would imply y - x + √2 is irrational.
- However, y - x + √2 simplifies to -(x - y + √2), which is the negation of x - y + √2.
- Since the negation of an irrational number is not necessarily irrational, the relation is not symmetric.
Transitive Property:
- For a relation to be transitive, if x is related to y and y is related to z, then x must be related to z.
- Consider xRy and yRz, which mean x - y + √2 and y - z + √2 are irrational.
- Now, (x - z) + 2√2 = (x - y + √2) + (y - z + √2) is irrational.
- Since 2√2 is irrational and the sum of two irrational numbers is not guaranteed to be irrational, the relation is not transitive.
Therefore, the relation R is reflexive as every real number is related to itself, but it is not symmetric or transitive.
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For real numbers x and y, we write xRy <=> x - y + √2 is an irrational number. Then, the relation R isa)reflexiveb)symmetricc)transitived)None of theseCorrect answer is option 'A'. Can you explain this answer?
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