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Symmetric Relations

Let A=1,2,3, ...

Question

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.

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Solution

$(1)$ R₁
Reflexivity:
Here,
$(1, 1), (2, 2), (3, 3) \in R So, R_{1} is reflexive .$

Symmetry:
$Here, (2, 1) \in R_{1}, but (1, 2) \notin R_{1} So, R_{1} is not symmetric .$

Transitivity:
$Here, (2, 1) \in R_{1} and (1, 3) \in R_{1}, but (2, 3) \notin R_{1} So, R_{1} is not transitive .$

$(2)$ R₂
Reflexivity:

$Clearly, (1, 1) and (3, 3) \notin R_{2} So, R_{2} is not reflexive .$

Symmetry:
$Here, (1, 3) \in R_{2} and (3, 1) \in R_{2} So, R_{2} is symmetric .$

Transitivity:
$Here, (1, 3) \in R_{2} and (3, 1) \in R_{2} But (3, 3) \notin R_{2} So, R_{2} is not transitive .$

$(3)$ R₃
Reflexivity:
$Clearly, (1, 1) \notin R_{3} So, R_{3} is not reflexive .$

Symmetry:
$Here, (1, 3) \in R_{3}, but (3, 1) \notin R_{3} So, R_{3} is not symmetric .$

Transitivity:
$Here, (1, 3) \in R_{3} and (3, 3) \in R_{3} Also, (1, 3) \in R_{3} So, R_{3} is transitive .$

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Similar questions

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. 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.

Q. Let A ={1, 2, 3} and let

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 wether or not each of the relations

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

on A is
(a) reflexive
(b) symmetric
(c) transitive

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}

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