Help with a simple proof.

It may be helpful to break down the proof and analyze the techniques at play for future insights.

In the exercise, it is given that f(x,y) ≤ f(y,x) for all (x,y) in R², so this can be utilized as a fact to move forward from. The goal is to show that for all (x,y) in R², f(x,y) = f(y,x).

Since the key words "for all" appear in the conclusion, the proof technique to use is to pick an arbitrary point (c,d) in R². The goal is to show that f(c,d) = f(d,c). If we can establish f(c,d) = f(d,c), then since (c,d) is an arbitrary point in R², it would follow that the statement must be true for all points (x,y) in R².

At this point,the key question to ask is how do we show that two objects are equal? One possible way to establish that two objects are equal is to show that objectA ≤ objectB and objectB ≤ objectA. Thus, if we show that f(c,d) ≤ f(d,c) and f(d,c) ≤ f(c,d), then f(c,d) = f(d,c) and the proof will be complete.

From here, to obtain the desired result, all one needs to do is observe that (c,d) and (d,c) are both in R² and f(x,y) ≤ f(y,x) for all (x,y) in R² and apply that fact to (c,d) and (d,c).

/r/learnmath Thread