3. For all the parts below, suppose that f : A B and let D...
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3. For all the parts below, suppose that f : A → B and let D be a subset of B. (a) Prove or give a counterexample: ƒ [ƒ−¹(D)] ≤ D. (b) Prove or give a counterexample: D ≤ ƒ [ƒ−¹(D)]. (c) State which of the following gives the "weakest" condition on ƒ to make a true statement. • Without any further conditions, it is true that ƒ [ƒ−¹(D)] = D. • If f is injective then ƒ [ƒ-¹(D)] = D. • If ƒ is surjective then ƒ [ƒ−¹(D)] = D. • If f is bijective then f [f-¹(D)] = D. Prove the statement that gives the weakest conditions on f. Note that the proof of your chosen statement can refer back to any proof you provide in parts (a) and (b) but a counterexample is not a proof. 3. For all the parts below, suppose that f : A → B and let D be a subset of B. (a) Prove or give a counterexample: ƒ [ƒ−¹(D)] ≤ D. (b) Prove or give a counterexample: D ≤ ƒ [ƒ−¹(D)]. (c) State which of the following gives the "weakest" condition on ƒ to make a true statement. • Without any further conditions, it is true that ƒ [ƒ−¹(D)] = D. • If f is injective then ƒ [ƒ-¹(D)] = D. • If ƒ is surjective then ƒ [ƒ−¹(D)] = D. • If f is bijective then f [f-¹(D)] = D. Prove the statement that gives the weakest conditions on f. Note that the proof of your chosen statement can refer back to any proof you provide in parts (a) and (b) but a counterexample is not a proof.
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a To prove or disprove ffD D we need to show that for every y in ffD y is also in D Lets ... View the full answer
Related Book For
Income Tax Fundamentals 2013
ISBN: 9781285586618
31st Edition
Authors: Gerald E. Whittenburg, Martha Altus Buller, Steven L Gill
Posted Date:
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