Suppose { A λ } , λ ∈ £ , represents a partition of the nonempty set A . Define R on A by x R y if and only if there is a subset A λ such that x ∈ A λ and y ∈ A λ . Prove that R is an equivalence relation on A and that the equivalence classes of R are the subsets A λ .
Suppose { A λ } , λ ∈ £ , represents a partition of the nonempty set A . Define R on A by x R y if and only if there is a subset A λ such that x ∈ A λ and y ∈ A λ . Prove that R is an equivalence relation on A and that the equivalence classes of R are the subsets A λ .
Solution Summary: The author demonstrates that R is an equivalence relation on a nonempty set A if the following conditions are satisfied.
Suppose
{
A
λ
}
,
λ
∈
£
, represents a partition of the nonempty set A. Define R on A by
x
R
y
if and only if there is a subset
A
λ
such that
x
∈
A
λ
and
y
∈
A
λ
. Prove that R is an equivalence relation on A and that the equivalence classes of R are the subsets
A
λ
.
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RELATIONS-DOMAIN, RANGE AND CO-DOMAIN (RELATIONS AND FUNCTIONS CBSE/ ISC MATHS); Author: Neha Agrawal Mathematically Inclined;https://www.youtube.com/watch?v=u4IQh46VoU4;License: Standard YouTube License, CC-BY