Let $R$ be a relation on $R$, given by $R=\{(a, b): 3 a-3 b+\sqrt{7}$ is an irrational number $\}$Then $R$ is
- reflexive and symmetric but not transitive
- reflexive and transitive but not symmetric
- reflexive but neither symmetric nor transitive
- an equivalence relation
The Correct Option is C
Approach Solution - 1
As
which belongs to relation so relation is reflexive
Check for symmetric:
Take
Now (a, b) but
As
which is rational so relation is not symmetric.
Check for Transitivity:
Take (a, b) as
as
So now (a, b) but which means relation is not transitive
Approach Solution -2
Check for reflexivity:
Since \[ 3(a - a) + \sqrt{7} = \sqrt{7} \] which is irrational, the relation is reflexive.
Check for symmetry:
Let \( a = \frac{\sqrt{7}}{3}, b = 0 \). Then \( (a, b) \in R \) but \( (b, a) \notin R \).
Since \[ 3(b - a) + \sqrt{7} = 0 \] which is rational, the relation is not symmetric.
Check for transitivity:
Let \( (a, b) = \left( 1, \frac{2\sqrt{7}}{3} \right) \) and \( (b, c) = \left( 1, \frac{2\sqrt{7}}{3} \right) \). Then, \( (a, b) \in R \) and \( (b, c) \in R \), but \( (a, c) \notin R \), so the relation is not transitive.
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Concepts Used:
Operations on Sets
Some important operations on sets include union, intersection, difference, and the complement of a set, a brief explanation of operations on sets is as follows:
1. Union of Sets:
- The union of sets lists the elements in set A and set B or the elements in both set A and set B.
- For example, {3,4} ∪ {1, 4} = {1, 3, 4}
- It is denoted as “A U B”
2. Intersection of Sets:
- Intersection of sets lists the common elements in set A and B.
- For example, {3,4} ∪ {1, 4} = {4}
- It is denoted as “A ∩ B”
3.Set Difference:
- Set difference is the list of elements in set A which is not present in set B
- For example, {3,4} - {1, 4} = {3}
- It is denoted as “A - B”
4.Set Complement:
- The set complement is the list of all elements present in the Universal set except the elements present in set A
- It is denoted as “U-A”