What did I think was difficult?
Fundamental Properties of Set Operations. I find that these actual properties are not too bad to understand when approached from a definition standpoint, but I struggle to implement them and completely grasp the concept when they are used in proofs.
I remember learning Commutative laws, Associative laws, distributive laws and De Morgan's Laws back in Stat 240. At the time, they were difficult for me to understand. Now, by themselves, they are a cinch. But when I have to use them in proofs, it's often difficult to remember to use them. With practice, I feel that I'll get it. The same thing with Cartesian products. I feel that I understand the basic examples of cartesian product proofs in the book, but I'm nervous for more complicated versions.
The section regarding contradiction was not too bad for me. In fact, I want to blog more about it now.
What did I think was interesting?
Counterexamples
Fundamental Properties of Set Operations. I find that these actual properties are not too bad to understand when approached from a definition standpoint, but I struggle to implement them and completely grasp the concept when they are used in proofs.
I remember learning Commutative laws, Associative laws, distributive laws and De Morgan's Laws back in Stat 240. At the time, they were difficult for me to understand. Now, by themselves, they are a cinch. But when I have to use them in proofs, it's often difficult to remember to use them. With practice, I feel that I'll get it. The same thing with Cartesian products. I feel that I understand the basic examples of cartesian product proofs in the book, but I'm nervous for more complicated versions.
The section regarding contradiction was not too bad for me. In fact, I want to blog more about it now.
What did I think was interesting?
Counterexamples
Wikipedia really helped clarify what a counterexample is.
If there is a general rule, say "all x are positive," you just need to find one case where x is negative in order to prove the proof wrong. This is especially useful with univeral quantifiers. I think I really like this because I see so often that people make statements that seem univeral. "Everyone hates me." "Everyone in my math class is confused." "No one likes him." It is dangerous to make such extreme examples! I think it's so dangerous because all you have to do is find a counterexample that can break your entire argument. I think when making a proof or argument, you should really think through the counterexamples that people present.
Counterexamples can be very powerful in proofs. Look for cases where you know the proof is false. Be creative in your thinking, and you'll find that the answer is often simple.
Thanks Themistocles.
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