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muAUB, AnB, A -B and B-A: 2 For Exercises (13) to (22): Prove the following Theorems concerning Real Numbers using only the 11 Field Axioms (and possibly Theorems that were proven in Chapter Zero). Specify in your proof which Axiom or Theorem you are using at each step. 13. Prove The Cancellation Law for Adition: For all x. y, c e R: Ifx + c=y+c, then x = y. 14. Prove The Cancellation Law for Multiplication : For all x, y, k E R, ks 0: lfk-x = k-y, then x *), Use The Multiplicative Property of Zero to prove that 0 cannot have a multiplicative inverse. Hint: Use Proof by Contradiction: Suppose 0 has a multiplicative inverse x.. Prove The Uniqueness of Additive Inverses: Suppose x e R. If w e R is any real number with the property that x + w 0 +x, then w-x. In other words, -x is the only real number that satisfies the above equations. 15. 16. 17. Use the previous Exercise to show that -0 0. Hint: which Field Axiom tells us what 0+0 is? 18. Use the Uniqueness of Additive Inverses to prove that for all x e R: - 19. 20. (1).x. Hint: simplify x (1).x. The Double Negation Property: Use some of the previous Exercises to show that: For all x e R (x)-x Prove The Uniqueness of Multiplicative Inverses: Suppose x e R and x # 0: If y E R is any real number with the property that x.y-1-y .x, then y 1/x. In other l/x is the only real number that satisfies the above equations. 19 16
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