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Ordered Pair

Determine whi...

Question

Determine which of the following are reflexive relations on set $A = (1, 2, 3}$ .
$R_{1} = ((1, 1), (2, 2), (3, 3), (2, 1)}$
$R_{2} = ((1, 1), (3, 3), (2, 1), (3, 2)}$ .
If $R_{1}$ is reflexive then answer $1$ and if $R_{2}$ is reflexive then answer $2$ .

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Solution

Given, $A = {1, 2, 3}$
Clearly, $(1, 1) (2, 2), (3, 3) \in R$
Thus, $R_{1}$ is a reflexive relation on set $A .$
But $R_{2}$ is not a reflexive relation on $A$ because $2 \in A$ but $(2, 2) \notin R_{2}$ .

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Similar questions

R_{1}

R_{2}

be two relations defined on a non empty set A. Which of the following statements is false? Give reasons in support of your answer.
(a) If

R_{1}

R_{2}

are reflexive, then so is

R_{1} \cap R_{2}

R_{1}

R_{2}

are symmetric, then so is

R_{1} \cup R_{2}

R_{1}

R_{2}

are transitive, then so is

R_{1} \cup R_{2}

R_{1}

R_{2}

are transitive, then so is

R_{1} \cap R_{2}

R_{1}

R_{2}

are symmetric, then so is

R_{1} \cap R_{2}

A = {1, 2, 3}

R_{1} = {(1, 1), (1, 3) (3, 1) (2, 2), (2, 1) (3, 3)}

R_{2} = {(2, 2), (3, 1), (1, 3)}

R_{3} = {(1, 3), (3, 3)}

R_{4} = A \times A .

Find which of the given each of the relations

R_{1}, R_{2}, R_{3}, R_{4},

on A satisfy all the given properties:

(a) reflexive (b) symmetric (c) transitive

Q. Let A = {1, 2, 3}, and let R₁ = {(1, 1), (1, 3), (3, 1), (2, 2), (2, 1), (3, 3)}, R₂ = {(2, 2), (3, 1), (1, 3)}, R₃ = {(1, 3), (3, 3)}. Find whether or not each of the relations R₁, R₂, R₃ on A is (i) reflexive (ii) symmetric (iii) transitive.

Q. Test whether the following relations R₁, R₂, and R₃ are (i) reflexive (ii) symmetric and (iii) transitive:
(i) R₁ on Q₀ defined by (a, b) ∈ R₁ ⇔ a = 1/b.
(ii) R₂ on Z defined by (a, b) ∈ R₂ ⇔ |a – b| ≤ 5
(iii) R₃on R defined by (a, b) ∈ R₃ ⇔ a²– 4ab + 3b²= 0.

Q. Three relations R₁, R₂ and R₃ are defined on a set A = {a, b, c} as follows:
R₁ = {(a, a), (a, b), (a, c), (b, b), (b, c), (c, a), (c, b), (c, c)}
R₂ = {(a, a)}
R₃ = {(b, c)}
R₄ = {(a, b), (b, c), (c, a)}.

Find whether or not each of the relations R₁, R₂, R₃, R₄ on A is (i) reflexive (ii) symmetric and (iii) transitive.

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