Let \(\mathscr{F}\) be the set of Pythagorean triples in \(\mathbb{N}^{3}=\mathbb{N} \times \mathbb{N} \times \mathbb{N}\). That is, if \((a, b, c) \in \mathbb{N} \times \mathbb{N} \times \mathbb{N}\) and \(a^{2}+b^{2}=c^{2}\) then \((a, b, c) \in \mathscr{T}\). Without
loss of generality assume that \(a \leq b \leq c\). Let \((a, b, c)\) and \((e, f, g) \in \mathscr{F} .\) Define \((a, b, c) R(d, e, f) \Leftrightarrow \frac{d}{a}=\frac{e}{b}=\frac{f}{c}\).
Note: If \((a, b, c) R(e, f, g)\) then \((a, b, c)\) and \((d, e, f)\) are considered to be the "same" triple. For example \((5,12,13) R(15,36,39)\) because \(\frac{15}{5}=\frac{36}{12}=\frac{39}{3}\).
Answer the following
(a) Show that \(R\) is an equivalence relation on \(\mathscr{T}\).
(b) Note that the following are Pythagorean triples: \((3,4,5),(5,12,13),(7,24,25),(9,40,41)\).
i. Find the next Pythagorean triple on this list, that is, \((11,-,-)\).
ii. Describe the pattern you see in 3 bi with a mathematical formula, that is, find the formula for the \(n\) th triple in the list \(\{(3,4,5),(5,12,13),(7,24,25),(9,40,41), \ldots\}\)
iii. Prove that the formula is true any way you would like.
iv. Is each triple in a different equivalence class? That is, can any two triples in the list \(\{(3,4,5),(5,12,13),(7,24,25),(9,40,41), \ldots\}\) be in the same equivalence class. Why would proving this be equivalent to showing that the number of distinct triples \(\mathrm{n}\) the list is infinite.
Homework Answers
Discrete Structures class. An ordered triple of positive integers (a,b,c) is called a Pythagorean Triple if...
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thank you very much
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Hello! I need help on the following problem. If possible, please
write explanations out in words so I can fully understand all the
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