This week we learned about how to prove a universal statement by using different methods and how to prove an existential statement.
In week 4 we proved a universal statement directly, but sometimes we'll find that prove some statements directly is not easy. In this case, we need to try some other methods to prove those statements. First, we know that a statement's contrapositive is equivalent to this statement. Thus, when we cannot prove a statement directly, we can try to prove the statement's contrapositive.
Another method to prove a statement is prove by contradiction. This method is similar to prove by using contrapositive. For example, the statement is A => B, to prove this by using contradiction, we can assume ¬B and if we get ¬A, then we can say that ¬B => ¬A and thus, A => B is correct.
To prove existential, what we need to do is to find an example, so that we can say there exist an element that can make the statement to be true. However, the difficult part here is how to choose a suitable example. Sometimes the example we choose may not be a constant like x = 2, the example may be related to some other parameters (i.e. "pick x = 2y").
this blog helps me understand contradiction and existence better, and it shows the method of how to do these proof.
回复删除This blog does introduce what we learned in the lecture. The blogger has perfectly understood the contents and has helped others comprehend them.
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