Consider the following statements in respect of any relation R on a set A : 1. If R is reflexive, then R-1 is also reflexive 2. If R is symmetric, then R-1 is also symmetric 3. If R is transitive, then R-1 is also transitive Which of the above statements are correct?
Consider the following statements in respect of any relation R on a set A : 1. If R is reflexive, then R-1 is also reflexive 2. If R is symmetric, then R-1 is also symmetric 3. If R is transitive, then R-1 is also transitive Which of the above statements are correct? Correct Answer 1, 2 and 3
Concept:
1). Reflexive: Each element is related to itself.
- R is reflexive if for all x ∈ A, xRx.
2). Symmetric: If any one element is related to any other element, then the second element is related to the first.
- R is Symmetric if for all x, y ∈ A, if xRy, then yRx.
3). Transitive: If any one element is related to a second and that second element is related to a third, then the first
element is related to the third.
- R is transitive if for all x, y, z ∈ A, if xRy and yRz, then xRz.
4). R is an equivalence relation if A is nonempty and R is reflexive, symmetric, and transitive.
5). Let R be a relation from a set A to another set B. Then R is of the form {(x, y): x ∈ A and y ∈ B}. The inverse
relationship of R is denoted by R-1 and its formula is R-1 = {(y, x): y ∈ B and x ∈ A}.
Calculation:
Statement I: If R is reflexive, then R-1 is also reflexive
R is reflexive
⇒ (a,a) ∈ R, a ∈ A
⇒ (a,a) ∈ R−1
⇒ R−1 is also reflexive relation.
Statement II: If R is symmetric, then R-1 is also symmetric
Let (b,a) ∈ R−1
⇒ (a,b) ∈ R, a,b ∈ A
⇒ (b,a) ∈ R
⇒ (a,b) ∈ R−1
If (b,a) ∈ R−1 then (a,b) ∈ R−1
⇒ R−1 is also symmetric relation.
Statement III: If R is transitive, then R-1 is also transitive
Let (b,a), (a,c) ∈ R−1
⇒ (a,b), (c,a) ∈ R
⇒ (c,a),(a,b) ∈ R
⇒ (c,b) ∈ R
⇒ (b,c) ∈ R-1
If (b,a), (a,c) ∈ R-1 then (b,c) ∈ R-1
⇒ R−1 is also transitive relation.
∴ R−1 is reflexive, symmetric and transitive.
