Let A and B be non-empty sets. Suppose f : A → B and g : A → B are bijections.Either prove the following are bijections, or find a counter-example: a) A = B = R and f + g : R → R is defined by (f + g)(x) = f(x) + g(x);b) A = B = R and fg: R → R is defined by (ƒg)(x) = f(x)g(x);c) Let ao E A. Then h: A → B × {a} is defined by h(a) = (f(a), ao).

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Answered: a) A = B = R and f+g: R → R is defined…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let A be a set and R and S be relations on A. Dene the relation R + S on A byR + S = (R-S) [ (S-R):Show that:(a) If A is countable, then so does R + S.(b) Show that the converse does not hold; that is, give an example of a set A that is uncountablebut R + S is countable.(c) Prove or disprove: If R and S are partial orders on A, then so does R + S.
Please answer with explanation... I was told my answer is not fully correct when selecting the 1st, third, and 4th option
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a) A = B = R and f+g: R → R is defined by (f + g)(x) = f(x) + g(x); b) A = B = R and fg: R → R is defined by (fg)(x) = f(x)g(x); c) Let ao E A. Then h: A → B × {a} is defined by h(a) = (f(a), ao).