Problem 26. The statement x y\\R is equivalent (using my notation) to the statem

Question

Problem 26. The statement x yR is equivalent (using my notation) to the statement x [y]. Both of them mean that x belongs to the equivalence class of y with respect to the equivalence relation R.

26. Statement. Theorem? Proof? A. y, z E A. Statement: Let R be an equivalence relation on And, let z, If r e y/R and 2 r/R, then 2 y/R. Suppose z E y/R, Proof? We prove this theorem by contradiction. is symmetric, z g r/R, and z E y/R. z E y/R, since R That is, TRy also. Since y/R, yRz, since R is transitive, rR2. z e T/R. This contradicts the hypothesis z I/R.

Explanation / Answer

Hello,

The proof given is fine.

What you expect from here?

Here in the proof if we assume that z belongs to y/R then we get z belongs to x/R which is not possbile since z is not related to x ( given).

Thus aur assumption that z R y is wrong.

Hence, z is not related to y.

Hence the proof.

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