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2.7 Prove that if theta:S-T, then theta(composition)identity=theta and identity(composition)theta=theta

2.20 Prove that the inverse of an invertible mapping is invertible

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2.7. Denote the identity mapping of S by i_{S} and the identity mapping in T by i_{T} (are not these two in the statement given in the text?). Now we have: q° i_{S}(x)=q(i_{S}(x)) by definition of composition; further q(i_{S}(x))=q(x) because i_{S}(x) = x; similarly i_{T}° q(x)) = q(x). So the mappings do the same thing to each x, and so they are equal.

2.20. It follows from the definition of inverse mapping that the inverse of the inverse of a is a. So, the inverse of a is invertible.