Analyzing the Relation R on the Interval (-1, 1)
We are given a relation R defined on the open interval \(A = (-1, 1)\). The relation is defined as \(R = \{(x, y) \in A \times A \mid |x + y| < 2\}\).
To determine the nature of this relation, we need to check if it satisfies the properties of reflexivity, symmetry, and transitivity. If all three properties hold, the relation is an equivalence relation.
Checking Reflexivity
A relation R on a set A is reflexive if for every element \(a \in A\), the pair \((a, a)\) is in R.
- We need to check if \((a, a) \in R\) for all \(a \in (-1, 1)\).
- The condition for \((a, a) \in R\) is \(|a + a| < 2\).
- This simplifies to \(|2a| < 2\).
- Which is equivalent to \(-2 < 2a < 2\).
- Dividing by 2, we get \(-1 < a < 1\).
- Since the relation is defined on the open interval \((-1, 1)\), this condition \(a \in (-1, 1)\) is true for all elements \(a\) in the set.
- Therefore, R is reflexive.
Checking Symmetry
A relation R on a set A is symmetric if whenever \((x, y) \in R\), then \((y, x) \in R\) for all \(x, y \in A\).
- Assume \((x, y) \in R\). This means \(x, y \in (-1, 1)\) and \(|x + y| < 2\).
- We need to check if \((y, x) \in R\). The condition for this is \(|y + x| < 2\).
- Since addition is commutative, \(x + y = y + x\).
- Therefore, \(|x + y| = |y + x|\).
- If \(|x + y| < 2\), it directly follows that \(|y + x| < 2\).
- Thus, if \((x, y) \in R\), then \((y, x) \in R\).
- Therefore, R is symmetric.
Checking Transitivity
A relation R on a set A is transitive if whenever \((x, y) \in R\) and \((y, z) \in R\), then \((x, z) \in R\) for all \(x, y, z \in A\).
- Assume \((x, y) \in R\) and \((y, z) \in R\). This means:
- \(x, y, z \in (-1, 1)\)
- \(|x + y| < 2\), which implies \(-2 < x + y < 2\).
- \(|y + z| < 2\), which implies \(-2 < y + z < 2\).
- We need to check if \((x, z) \in R\), which means we need to verify if \(|x + z| < 2\).
- Consider the properties of the interval \(A = (-1, 1)\). For any elements \(x, z \in (-1, 1)\):
- We know \(x < 1\) and \(z < 1\). Adding these inequalities gives \(x + z < 1 + 1\), so \(x + z < 2\).
- We also know \(x > -1\) and \(z > -1\). Adding these inequalities gives \(x + z > -1 + (-1)\), so \(x + z > -2\).
- Combining these, we have \(-2 < x + z < 2\) for all \(x, z \in (-1, 1)\).
- This inequality is precisely the condition \(|x + z| < 2\).
- Therefore, for any \(x, z \in (-1, 1)\), the pair \((x, z)\) is always in R, regardless of the value of y, as long as \(x, y, z\) are in the set \((-1, 1)\).
- Since the condition for \((x, z) \in R\) is automatically satisfied for any \(x, z\) in the base set \((-1, 1)\), the transitivity property holds.
- Therefore, R is transitive.
Conclusion
The relation R is:
- Reflexive
- Symmetric
- Transitive
A relation that is reflexive, symmetric, and transitive is called an equivalence relation.
Thus, the relation R is an equivalence relation.
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 : - 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 : - 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 \(y \in B\)?
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