\(R = \{(x, y) : x, y \in N \text{ and } x = y^3\}\).
Which of the following statements is/are correct ?
I. R is symmetric relation
II. R is transitive relation
Select the answer using the code given below :
To determine if the relation \( R \) on the set of natural numbers \( \mathbb{N} \), defined as \( R = \{(x, y): x, y \in \mathbb{N} \text{ and } x = y^3\} \), is symmetric and transitive, let's analyze each property step-by-step:
- Symmetric Relation: A relation \( R \) is symmetric if for all \( (x, y) \in R \), \( (y, x) \in R \) as well. Let's test this for our relation:
- Given \( (x, y) \in R \), it implies \( x = y^3 \).
- To satisfy the symmetric property, \( (y, x) \) should also belong to \( R \), which means \( y = x^3 \).
- However, if \( x = y^3 \) and \( y = x^3 \) then, by substitution, we have \( y^3 = (y^3)^3 \). Simplifying, \( y^9 = y^3 \). This implies \( y = 0 \) or \( y = 1 \) for trivial solutions, but since \( 0 \not\in \mathbb{N} \), only \( y = 1 \) is valid.
- Thus, the relation is not symmetric for all \( x, y \in \mathbb{N} \).
- Transitive Relation: A relation \( R \) is transitive if for all \( (x, y), (y, z) \in R \), then \( (x, z) \in R \). Let's check this for our relation:
- Suppose \( (x, y) \in R \implies x = y^3 \), and \( (y, z) \in R \implies y = z^3 \).
- For transitivity, we need \( (x, z) \in R \), which means \( x = z^3 \).
- Substituting the equations: if \( y = z^3 \), then substituting in \( x = y^3 \) gives \( x = (z^3)^3 = z^9 \), which is not equal to \( z^3 \) unless \( z = 1 \).
- Hence, the relation is not transitive for all \( x, y, z \in \mathbb{N} \).
Since the relation \( R \) is neither symmetric nor transitive, the correct answer is:
Neither I nor II
Similar Questions
- Let P and Q be two non-void relations on a set A. Which of the following statements are correct?
I. P and Q are reflexive \(\Rightarrow P \cap Q\) is reflexive.
II. P and Q are symmetric \(\Rightarrow P \cup Q\) is symmetric.
III. P and Q are transitive \(\Rightarrow P \cap Q\) is transitive.
Select the answer using the code given below. - Let \(A = \{1, 2, 3, 4, 5\}\) and \(B = \{6, 7\}\). What is the number of onto functions from \(A\) to \(B\)?
- Let \(A = \{x \in R: -1 > x < 1\}\). Which of the following is/are bijective functions from \(A\) to itself?
1. \(f(x) = x|x|\)
2. \(g(x) = \cos(\pi x)\)
Select the correct answer using the code given below : - Let \(R\) be a relation on the open interval \((-1, 1)\) and is given by \(R = \{(x, y) : |x + y| < 2\}\). Then which one of the following is correct?
- If \(f(x) = ax-b\) and \(g(x) = cx + d\) are such that \(f(g(x)) = g(f(x))\), then which one of the following holds ?
- Let \(A = \{-3,-2,-1, 0, 1, 2, 3\}\) and \(B = \{0, 1, 4, 9\}\). How many elements does the subset of \(A \times B\) corresponding to the relation \(R = \{(x, y): |x| < y\}\) have, where \(x \in A\) and \(y \in B\)?
Important Questions from Relations and Functions
If the function of \(f(x)=\dfrac{x}{x-1}\) express f(3x) in terms of f(x)
If \(f(x)-\dfrac{1}{1+2^{1/x}}\) then at x = 0 the function is:
If f(x) is a periodic function and a is a positive real number such that f(x + 2α) + f(x) = 0 for all x ∈ ℝ, then the period of f(x) is:
If \(f(x)=\frac{1}{1+x}\), g(x) = f{f(x)} and h(x) = f[f{f(x)}], then the value of f(x).g(x).h(x) is:
If \(f(x) = {\sin ^{ - 1}}\left[ {\frac{{\sqrt 3 }}{2}x - \frac{1}{2}\sqrt {1 - {x^2}} } \right]\), \(x \in \left[ { - \frac{1}{2},1} \right]\), then f(x) will be
