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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Can you use set theory concepts , like the multiplication operation to solve problem 1 please
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.)
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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