(a) Let n be a non-negative integer. Prove that if n is a multiple of 3, then n² is a multiple of 3. (b) Let n be a non-negative integer. Prove that if n is a not a multiple of 3, then n² is 1 more than a multiple of 3. (c) Let T(n) be the predicate "n is a multiple of 3", and let S(n) be the predicate "n² is a multiple of 3". Taking the domain for n to be the non-negative integers, determine whether the following propositions are true or false, and give proofs of your assertion \n (T(n) ⇒ S(n)). En (T(n) ^-S(n)). Vn (S(n) ⇒T(n)). Vn (S(n) T(n)). You may of course refer to anything you proved in earlier parts without having to copy over the proof.
(a) Let n be a non-negative integer. Prove that if n is a multiple of 3, then n² is a multiple of 3. (b) Let n be a non-negative integer. Prove that if n is a not a multiple of 3, then n² is 1 more than a multiple of 3. (c) Let T(n) be the predicate "n is a multiple of 3", and let S(n) be the predicate "n² is a multiple of 3". Taking the domain for n to be the non-negative integers, determine whether the following propositions are true or false, and give proofs of your assertion \n (T(n) ⇒ S(n)). En (T(n) ^-S(n)). Vn (S(n) ⇒T(n)). Vn (S(n) T(n)). You may of course refer to anything you proved in earlier parts without having to copy over the proof.
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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Transcribed Image Text:(a) Let n be a non-negative integer. Prove that if n is a multiple of 3, then n² is a multiple
of 3.
(b) Let n be a non-negative integer. Prove that if n is a not a multiple of 3, then n² is 1
more than a multiple of 3.
(c) Let T(n) be the predicate “n is a multiple of 3", and let S(n) be the predicate “n² is
a multiple of 3". Taking the domain for n to be the non-negative integers, determine
whether the following propositions are true or false, and give proofs of your assertion
Vn (T(n) ⇒ S(n)).
En (T(n) ^-S(n)).
Vn (S(n)=
→ T(n)).
Vn (S(n)T(n)).
You may of course refer to anything you proved in earlier parts without having to copy
over the proof.
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