What was difficult for me?
trivial proof - "a trivial proof refers to a statement involving a material implication where the consequent, or Q, in P→Q, is always true"
vacuous proof - "a vacuous truth is a statement that asserts that all members of the empty set have a certain property."
The concept of a vacuous proof was difficult for me. I at first didn't understand how it worked. So I did what I always do when I don't understand something - I went to the internet. I found this example on wikipedia that helped make so much sense of this. "For example, the statement "all cell phones in the room are turned off" may be true simply because there are no cell phones in the room. In this case, the statement "all cell phones in the room are turned on" would also be true, and vacuously so, as would the conjunction of the two: "all cell phones in the room are turned on and turned off"."
So despite the original difficulty in understanding vacuous proofs, I figured it out. My understanding is that when the "if" part of the implication statement is always false, then no matter when the second part of the implication is, the overall implication is going to be true by the truth values in the truth tables. I could see this being helpful later on down the road.
What did I find interesting?
From this reading, I really enjoyed reading through the "thought process" of the proof analysis. I loved the introduction of proof strategy. It aays this from the book - "from time to time, we will find ourselves in a position where we have a result to prove and it may not be entirely clear how to proceed. In such a case, we need to consider our options and develop a plan." This is called a proof strategy. I like this a lot. Recently, I read a book called Lone Survivor by a Navy Seal who was a part of an op that went wrong. They didn't know what to do. But they weighed their options, developed a plan, and moved forward. We cannot always be paralyzed by fear. We must learn how to move forward. I think that many times I have proofs to solve, and I get so confused and scared, that I freeze up. With a better proof strategy, I think I can be more effective in tackling these proofs. I enjoyed reading through the different proof analyses, and the way the book tackled these proofs. With practice, I can see myself becoming quite proficient at these proofs.
trivial proof - "a trivial proof refers to a statement involving a material implication where the consequent, or Q, in P→Q, is always true"
vacuous proof - "a vacuous truth is a statement that asserts that all members of the empty set have a certain property."
The concept of a vacuous proof was difficult for me. I at first didn't understand how it worked. So I did what I always do when I don't understand something - I went to the internet. I found this example on wikipedia that helped make so much sense of this. "For example, the statement "all cell phones in the room are turned off" may be true simply because there are no cell phones in the room. In this case, the statement "all cell phones in the room are turned on" would also be true, and vacuously so, as would the conjunction of the two: "all cell phones in the room are turned on and turned off"."
So despite the original difficulty in understanding vacuous proofs, I figured it out. My understanding is that when the "if" part of the implication statement is always false, then no matter when the second part of the implication is, the overall implication is going to be true by the truth values in the truth tables. I could see this being helpful later on down the road.
What did I find interesting?
From this reading, I really enjoyed reading through the "thought process" of the proof analysis. I loved the introduction of proof strategy. It aays this from the book - "from time to time, we will find ourselves in a position where we have a result to prove and it may not be entirely clear how to proceed. In such a case, we need to consider our options and develop a plan." This is called a proof strategy. I like this a lot. Recently, I read a book called Lone Survivor by a Navy Seal who was a part of an op that went wrong. They didn't know what to do. But they weighed their options, developed a plan, and moved forward. We cannot always be paralyzed by fear. We must learn how to move forward. I think that many times I have proofs to solve, and I get so confused and scared, that I freeze up. With a better proof strategy, I think I can be more effective in tackling these proofs. I enjoyed reading through the different proof analyses, and the way the book tackled these proofs. With practice, I can see myself becoming quite proficient at these proofs.
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