Mistakes regarding proofs: Difference between revisions
(Created page with "I had a teacher who would repeat many times in different classes '''"If you write 0 = 0 I'm going to give you a zero in the exam!"'''. Suppose we want to prove that <math>a + b = c + d</math>. We begin by saying that <math>a + b = x</math>. Then we say that <math>c + d = x</math>. Therefore <math>x = x \iff a + b = c + d</math>. Eureka! We didn't prove anything! What is wrong in the reasoning that we just did? The mistake is that we assumed that the equation is true wit...") |
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I had a teacher who would repeat many times in different classes '''"If you write 0 = 0 I'm going to give you a zero in the exam!"'''. | I had a teacher who would repeat many times in different classes '''"If you write 0 = 0 I'm going to give you a zero in the exam!"'''. | ||
Suppose we want to prove that <math>a + b = c + d</math>. We begin by saying that <math>a + b = x</math>. Then we say that <math>c + d = x</math>. Therefore <math>x = x \iff a + b = c + d</math>. Eureka! We didn't prove anything! What is wrong in the reasoning that we just did? The mistake is that we assumed that the equation is true without knowing whether it holds or not. We can't do such assumptions! | Suppose we want to prove that <math>a + b = c + d</math>. We begin by saying that <math>a + b = x</math>. Then we say that <math>c + d = x</math>. Therefore <math>x = x \iff a + b = c + d</math>. Eureka! We didn't prove anything! What is wrong in the reasoning that we just did? The mistake is that we assumed that the equation is true without knowing whether it holds or not. We can't do such assumptions! That why in linear algebra and calculus many properties are true if we restrict the conditions. Other times we are presented with counter-examples to show that some property is true for some cases, but not for all of them. | ||
Revision as of 03:35, 29 January 2022
I had a teacher who would repeat many times in different classes "If you write 0 = 0 I'm going to give you a zero in the exam!".
Suppose we want to prove that [math]\displaystyle{ a + b = c + d }[/math]. We begin by saying that [math]\displaystyle{ a + b = x }[/math]. Then we say that [math]\displaystyle{ c + d = x }[/math]. Therefore [math]\displaystyle{ x = x \iff a + b = c + d }[/math]. Eureka! We didn't prove anything! What is wrong in the reasoning that we just did? The mistake is that we assumed that the equation is true without knowing whether it holds or not. We can't do such assumptions! That why in linear algebra and calculus many properties are true if we restrict the conditions. Other times we are presented with counter-examples to show that some property is true for some cases, but not for all of them.
