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State whether the following statements are TRUE or FALSE:

The union of two equivalence relations is also an equivalence relation.
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4 Answers

Best answer
31 votes
31 votes

No union of two equivalence relations may not be an equivalence relation because of transitive dependency.

Equivalence relation: satisfy Reflexive, symmetric, and transitive property

  • Reflexive $\cup$ Reflexive = Reflexive
  • Symmetric $\cup$ Symmetric = Symmetric
  • Transitive $\cup$ Transitive $\neq$ Transitive why

Example $: R = \left \{ \left ( 1,2 \right ), \left ( 3,4 \right ), \left ( 1,4 \right ) \right \},\;S$ $=$ ${ (2, 3)}$

Union $= \left \{ \left ( 1,2 \right ), \left ( 3,4 \right ), \left ( 1,4 \right ), (2, 3) \right \}$  which is not transitive i.e. $(1, 3)$  and $(2, 4)$ is missing.

So, False is the answer.

edited by
15 votes
15 votes

https://math.stackexchange.com/questions/117285/symmetry-and-transitivity-for-the-union-of-two-equivalence-relations

Transitivity fails.

Let $A$=$ {1,2,3} $, $R$= $ {(1,1),(2,2),(3,3),(1,2),(2,1)} $ and $S$=$ {(1,1),(2,2),(3,3),(2,3),(3,2)} $, then $R∪S$

contains both $(1,2)$ and $(2,3)$, but not $(1,3)$

The symmetry could be worded better, but is alright. The important thing is that if (x,y)∈R∪S

then $(x,y)∈R$ or $(x,y)∈S$, but both $R$ and $S$ are symmetrical so $(y,x)$ must be contained in at least one of them (here I

use "or" instead of "and" you have used).

6 votes
6 votes

Union of two equivalence relation may or may not be an equivalence relation because it might not be transitive always.Though intersection of two equivalence relation is always an equivalence relation.

So, the given statement The union of two equivalence relations is also an equivalence relation = FALSE
0 votes
0 votes

Short trick

----------------------------------

  • Reflexive ∪ Reflexive = Reflexive
  • Symmetric ∪ Symmetric = Symmetric
  • Transitive ∪ Transitive ≠ Transitive 
  •  equivalence relations ∪ equivalence relations  ≠ equivalence relations
  •  equivalence relations ∩   equivalence relations = equivalence relations

hope my answer helps to solve more easily 

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