15
To solve the problem, we need to determine the number of elements in the subset of \(A \times B\) that satisfy the relation \(R = \{(x, y): |x| < y\}\) where \(x \in A\) and \(y \in B\).
First, let's break down the problem:
- \(A = \{-3, -2, -1, 0, 1, 2, 3\}\)
- \(B = \{0, 1, 4, 9\}\)
The condition \(R = \{(x, y): |x| < y\}\) means that for every pair \((x, y)\), the absolute value of \(x\) must be less than \(y\).
Let's check each \(y\) in set \(B\) and find suitable \(x\) that satisfy the relation:
- For \(y = 0\)
- No values of \(x\) satisfy \(|x| < 0\). Hence, no suitable pairs.
- For \(y = 1\)
- Satisfies for \(x = 0\). (1 pair: (0, 1))
- For \(y = 4\)
- Satisfies for \(x = -3, -2, -1, 0, 1, 2, 3\). (7 pairs: (-3, 4), (-2, 4), (-1, 4), (0, 4), (1, 4), (2, 4), (3, 4))
- For \(y = 9\)
- Satisfies for all \(x = -3, -2, -1, 0, 1, 2, 3\). (7 pairs: (-3, 9), (-2, 9), (-1, 9), (0, 9), (1, 9), (2, 9), (3, 9))
To find the total number of elements in the relation \(R\), we sum up the pairs found:
- Pairs with \(y = 0\): 0
- Pairs with \(y = 1\): 1
- Pairs with \(y = 4\): 7
- Pairs with \(y = 9\): 7
Total number of pairs: \(0 + 1 + 7 + 7 = 15\).
Thus, the correct number of elements in the subset is 15.
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 R be a relation on the set N of natural numbers defined by
\(R = \{(x, y) : x, y \in N \text{ and } x = y^3\}\).
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 :
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
