3.1.7 Study Project: Uniqueness of identities and inverses 1 Study Project: Uniqueness of identities and inverses The algebraic axioms of the real number system declare the existence of an additive identity, 0, and a multiplicative identity, 1. Uniqueness of these identities is not stated in the axioms because this can be proven from the axioms. Similarly, the axioms declare the existence of additive and multiplicative inverses but not the uniqueness of inverses since this also follows from the axioms. 1. Uniqueness of the additive identity a) An algebraic axiom of states that there exists an additive identity, 0. We wish to show that 0 is the only additive identity. Suppose another real number acts as an additive identity: for all x ∈ . In each step of the proof below, state either one axiom or the hypothesis as justification. Prove that the additive identity in is unique. Hypothesis: Suppose ∃ e ∈ such that for all x ∈ . Conclusion: Show that . Proof: Reason: Reason: Reason: b) Suppose there exist real numbers and such that . How does this hypothesis compare to the hypothesis in (a)? Is this hypothesis enough to conclude that must be the additive identity 0? In each step of the proof below, state either one axiom or the hypothesis as justification. Hypothesis: Let a ∈ , and suppose ∃ e ∈ such that a + e = a . Conclusion: Show that . Proof: Reason: Reason: Reason: Reason: Reason: Reason: Barbara A. Shipman, Active Learning Materials for a First Course in Real Analysis www.uta.edu/faculty/shipman/analysis. Supported in part by NSF grant DUE-‐0837810 3.1.7 Study Project: Uniqueness of identities and inverses 2 2. Uniqueness of the multiplicative identity a) An algebraic axiom of states that there exists a multiplicative identity, 1. We wish to show that 1 is the only multiplicative identity. State the hypothesis and the conclusion in proving that the multiplicative identity in is unique. Then, using the style of proof in Part (1), prove your conclusion in a sequence of steps, justifying each step by either one algebraic axiom of or the hypothesis. b) Suppose there exist real numbers and such that . How does this hypothesis compare to the hypothesis in (a)? Is this enough to conclude that must be the multiplicative identity 1? Either provide a counterexample or prove your conclusion in a sequence of steps, justifying each step by either one algebraic axiom of or the hypothesis. 3. Uniqueness of additive inverses Given a ∈ , an axiom states that there exists a real number such that . We wish to show that is the only additive inverse of . Suppose another real number acts as an additive inverse of , that is, suppose that . We must show that . In each step of the proof below, state either one axiom or the hypothesis as justification. Prove that additive inverses in are unique: Hypothesis: Let a ∈ , and suppose ∃ b ∈ such that . Conclusion: Show that . Proof: Reason: Reason: Reason: Reason: Reason: Reason: Barbara A. Shipman, Active Learning Materials for a First Course in Real Analysis www.uta.edu/faculty/shipman/analysis. Supported in part by NSF grant DUE-‐0837810 3.1.7 Study Project: Uniqueness of identities and inverses 3 4. Uniqueness of multiplicative inverses Given a ∈ with , an axiom states that there exists a real number such that . We wish to show that is the only multiplicative inverse of . Referring to Problem 3 as an example, state the hypothesis and the conclusion in proving that multiplicative inverses in are unique. Then prove your conclusion in a sequence of steps, justifying each step by either one algebraic axiom of or the hypothesis. 5. Which axioms are needed? a) In proving that additive and multiplicative identities are unique, which of the axioms of (, +, i) are used? Are there any that are not needed? b) Are all of the axioms of (, +, i) necessary in proving that additive and multiplicative inverses are unique? ■ Barbara A. Shipman, Active Learning Materials for a First Course in Real Analysis www.uta.edu/faculty/shipman/analysis. Supported in part by NSF grant DUE-‐0837810
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