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Exercise 1. Prove that m 1 = m. (Hint: it is not necessary to argue by induction on m.) Exercise 2. Prove the following statements: (a) If A is a set, then A is transitive iff A C P(A). (b) If A is a transitive set, then P(A) is a transitive set. (c) If B is a set of transitive sets, then UB is a transitive set. Exercise 3. Prove that for all m Ew, either m = Ø or ØЄ m. (Hint: Show that S = {m Є w | m = 0 or 0 Є m} is inductive.) The Multiplication Operation Next we will define the multiplication operation recursively in terms of the addition operation so that: m.o = = 0 m.n+ = m.n+m • For each mЄw, the Recursion Theorem gives a unique function Mm: ww such that Mm(0) Mm(n+) = 0 = Mm(n) +m As above, we can define the multiplication operation by = = M(m, n) Mm(n). Simon Thomas (Rutgers University) Math 361 The Multiplication Operation September 12, 2024 Notation From now on, we will write m. n = M(m, n). ⚫ Then the defining equations for multiplication can be rewritten as: m.0 = m.n+ = 0 m.n+m

Exercise 1. Prove that m 1 = m. (Hint: it is not necessary to argue by induction on m.) Exercise 2. Prove the following statements: (a) If A is a set, then A is transitive iff A C P(A). (b) If A is a transitive set, then P(A) is a transitive set. (c) If B is a set of transitive sets, then UB is a transitive set. Exercise 3. Prove that for all m Ew, either m = Ø or ØЄ m. (Hint: Show that S = {m Є w | m = 0 or 0 Є m} is inductive.) The Multiplication Operation Next we will define the multiplication operation recursively in terms of the addition operation so that: m.o = = 0 m.n+ = m.n+m • For each mЄw, the Recursion Theorem gives a unique function Mm: ww such that Mm(0) Mm(n+) = 0 = Mm(n) +m As above, we can define the multiplication operation by = = M(m, n) Mm(n). Simon Thomas (Rutgers University) Math 361 The Multiplication Operation September 12, 2024 Notation From now on, we will write m. n = M(m, n). ⚫ Then the defining equations for multiplication can be rewritten as: m.0 = m.n+ = 0 m.n+m

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.2: Exponents And Radicals
Problem 91E
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Can you use set theory concepts , like the multiplication operation to solve problem 1 please

Exercise 1. Prove that m 1 = m. (Hint: it is not necessary to argue by induction on m.) Exercise 2. Prove the following statements: (a) If A is a set, then A is transitive iff A C P(A). (b) If A is a transitive set, then P(A) is a transitive set. (c) If B is a set of transitive sets, then UB is a transitive set. Exercise 3. Prove that for all m Ew, either m = Ø or ØЄ m. (Hint: Show that S = {m Є w | m = 0 or 0 Є m} is inductive.)
Transcribed Image Text:Exercise 1. Prove that m 1 = m. (Hint: it is not necessary to argue by induction on m.) Exercise 2. Prove the following statements: (a) If A is a set, then A is transitive iff A C P(A). (b) If A is a transitive set, then P(A) is a transitive set. (c) If B is a set of transitive sets, then UB is a transitive set. Exercise 3. Prove that for all m Ew, either m = Ø or ØЄ m. (Hint: Show that S = {m Є w | m = 0 or 0 Є m} is inductive.)
The Multiplication Operation Next we will define the multiplication operation recursively in terms of the addition operation so that: m.o = = 0 m.n+ = m.n+m • For each mЄw, the Recursion Theorem gives a unique function Mm: ww such that Mm(0) Mm(n+) = 0 = Mm(n) +m As above, we can define the multiplication operation by = = M(m, n) Mm(n). Simon Thomas (Rutgers University) Math 361 The Multiplication Operation September 12, 2024 Notation From now on, we will write m. n = M(m, n). ⚫ Then the defining equations for multiplication can be rewritten as: m.0 = m.n+ = 0 m.n+m
Transcribed Image Text:The Multiplication Operation Next we will define the multiplication operation recursively in terms of the addition operation so that: m.o = = 0 m.n+ = m.n+m • For each mЄw, the Recursion Theorem gives a unique function Mm: ww such that Mm(0) Mm(n+) = 0 = Mm(n) +m As above, we can define the multiplication operation by = = M(m, n) Mm(n). Simon Thomas (Rutgers University) Math 361 The Multiplication Operation September 12, 2024 Notation From now on, we will write m. n = M(m, n). ⚫ Then the defining equations for multiplication can be rewritten as: m.0 = m.n+ = 0 m.n+m
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