MHB When to use equivalence relations? How to write it in octave?

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The discussion centers on the concept of equivalence in mathematical expressions, particularly in the context of the rot90 function in Octave. The rot90 function is used to rotate a matrix by a specified angle in multiples of 90 degrees. The example provided illustrates that rotating a matrix by -90 degrees is equivalent to rotating it by 270 degrees (or 3 times 90 degrees), as well as by 630 degrees (or 7 times 90 degrees). This demonstrates that certain expressions can yield identical results despite different inputs, indicating an equivalence relation among them. The discussion emphasizes that the equivalence relation is formed by the set of all pairs of expressions that produce the same outcome, regardless of the specific angle used for rotation. Understanding this concept is crucial for working with matrix transformations in Octave.
WMDhamnekar
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Sometimes to help describe one expression, another expression is shown that produces identical results. The exact equivalence of expressions is indicated with ‘ ≡’.

For example: rot90 ([1, 2; 3, 4], -1) ≡ rot90 ([1, 2; 3, 4], 3) ≡ rot90 ([1, 2; 3, 4], 7)

What is the meaning of 'rot90;?

What is the meaning of this example?

How to write equivalence relation in octave?

How does all of the above expressions have equivalence relation?
 
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I don't know Octave, but rot90(x, k) probably is the result of rotating x by $k\cdot90^\circ$. I am not sure what [1, 2; 3, 4] represents: a matrix, points coordinates or something else, but this may not be important in this example. The important point is that rotating by $3\cdot90^\circ$ is the same as rotating by $7\cdot90^\circ$, which is also the same as rotating by $90^\circ$ in the opposite direction. Therefore, expressions rot90 ([1, 2; 3, 4], -1), rot90 ([1, 2; 3, 4], 3) and rot90 ([1, 2; 3, 4], 7) are equivalent. The set of all pairs of equivalent expressions forms an equivalence relation.
 

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