3.1.6 Study Project: Distributive properties 1 Study Project: Distributive properties Recall the distributive property of multiplication over addition, one of the nine algebraic axioms of listed in Supplement 3.1.1: For all x, y, z in , . In this study, we are interested in distributive properties involving other pairs of operations, including subtraction and division, defined as follows: Division: For Subtraction: For with , , . . 1. Multiplication over subtraction a) According to your experience with algebraic operations, is multiplication distributive over subtraction? Write a precise mathematical expression of this, and decide whether you believe it is true or false. b) Consider the following argument that multiplication is distributive over subtraction. Is each step correct? Can each step be justified by either one axiom of or the definition of subtraction? Multiplication is distributive over subtraction: Hypothesis: Let . Conclusion: Show that . Proof: Reason: Reason: Reason: Reason: c) In Part (b), we discovered that we need a lemma stating that to complete the proof that multiplication is distributive over subtraction. Fill in the proof of this lemma below. Lemma 1: Hypothesis: Let . Conclusion: Show that . 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 3.1.6 Study Project: Distributive properties 2 Proof: Reason: Additive identity Reason: Reason: Associativity of + Reason: Distributive axiom Reason: Additive inverses Reason: Reason: Can each step be justified by either one axiom of or the definition of subtraction? If not, what lemma will we need to complete the proof of Lemma 1? d) Complete the proof Lemma 2 below, showing that for , , where each step is justified by one axiom of . Hypothesis: Let . Conclusion: Show that . Proof: Reason: Additive identity Reason: Additive inverses Reason: Multiplicative identity Reason: Associativity of + Reason: Distributive axiom Reason: Commutativity of + Reason: Additive identity Reason: Multiplicative inverses Reason: (e) Give a summary of how we have proved that multiplication is left distributive over subtraction using only the algebraic axioms of (, +, i) and the definition of subtraction. (f) Is multiplication also right distributive over subtraction? Explain why we may simple state that multiplication is distributive over subtraction. 2. Division over addition a) Is division left distributive over addition? What equality must be shown to establish this? Prove your answer. b) Prove that division is right distributive over addition using only the algebraic axioms of (, +, i) and the definition of division. 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 3.1.6 Study Project: Distributive properties 3 3. Division over subtraction Is division left distributive over subtraction? Is division right distributive over subtraction? Prove your answers. 4. Other distributive properties? Taking combinations of the four operations and are there any other distributive properties that you are able to state? Consider some more possibilities and explain your results. 5. Summary Report your findings on distributive properties of and in a clear, complete, and organized way, as you would like it to appear in an official handbook on properties of operations. ■ 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