Math CentralQuandaries & Queries


Question from nazrul, a teacher:

If p^2-pa+q=0 and m^2-ma+n=0 , can we write p^2-pa+q= m^2-ma+n? Please explain.


Yes, because the equality relation is symmetric
(a=b implies b=a) and transitive (a=b and b=c implies a=c). As Euclid put it, things equal to the same thing are equal.

But note that neither of the first two expressions is true for ALL values of p,a,q,n ; they are not tautologies like

x2 -a2 = (x+a)(x-a),

but "predicates": statements about the variables which may be true or false. In order to be able to draw the conclusion you do, you must have values in mind for p,a,q,n for which both the original equations hold.

Good Hunting!

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