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Question: 14...

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IU. llJ() is continuous on the closed interval [a,b] then f(x) possesses both a maxi minimum on [a, b] For Exercises (11) and (12): For the sets A and B, find AUB, AnB, A -B and B 11. A (a,cf,h,ij,m), B b,c,g,hj,p.q 12. . A = (a,d,g,h.jp.r4), B (b,d,g,h,k,p,q,s,t,v} For Exercises (13) to (22): Prove the following Theorems concerning Real Numbers the 11 Field Axioms (and possibly Theorems that were proven in Chapter Zero). Spe proof which Axiom or Theorem you are using at each step. 13. Prove The Cancellation Law for Addition: For all x, y, c E R: 14. Prove The Cancellation Law for Multiplication: For all x, y, k e R, k丰0: 15. Use The Multiplicative Property of Zero to prove that 0 cannot have a multiplicativ 16. Prove The Uniqueness of Additive Inverses: Suppose xe R. Ifw R is any real nu Ifx + c=y+c, then x=y. If k.x -k.y, then x y. Hint: Use Proof by Contradiction: Suppose 0 has a multiplicative inverse x... the property that x+w 0 w+x, then w-x. In other words, -x is the only real nu satisfies the above equations. 17. Use the previous Exercise to show that-0 = 0. Hint: which Field Axiom tells us what 18. Use the Uniqueness of Additive Inverses to prove that for all x e R:- 1) .x Hint: simplifyx+(1).x. Dsonout.. IIse some of the nrevious Exercises to show that: For a 14
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