Question

2.3. Prove that:
The inverse of an equivalence relation is also an equivalence
relation​

Answer 1

Proved That:

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• Let R be an equivalence relation.

⠀Reflexive (a,a) R aA

⠀Symmetric (a,b) R

⇒ (b,a) R

⠀Transitive (a,b), (b,c) R

⇒ ac R

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• Let R-1 is the inverse relation of R

⠀Now R-1 is reflexive

⇒ (a,a) R

⇒ (a,a) R-1

• Symmetric

⠀(b,a) R ⇒ (a,b) R-1

⠀(a,b) R ⇒ (b,a) R-1

⠀(a,b), (b,a) R-1

⠀So R-1 is symmetric.

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• Transitive

⠀(a,b), (b,c), (a,c) R

⠀Now, (b,a) R-1 (a,c) R-1

⠀(c,b) R-1

⠀Since R-1 is symmetric (b,c) R-1

⠀So, R-1 is transitive.

⠀Hence R-1 is an equivalence relation.

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Proved✔

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