Discrete Mathematics: Proof by Contradiction and Contrapositive
Reply Requirements
Within your replies to your classmates, compare your posts and discuss the following:
Is the proof well-written and logically clear and correct?
Explain how the type of proof that was not chosen could or could not be used for the same example.
Your replies to other student posts are expected to be mathematically substantive, relevant, original, and engaging. What does this mean? Remember that these replies should be in a form where you are engaging in communications at the college level. If your replies sound like the type of communication that one would encounter within an informal chat or chat room, it is most likely inappropriate. Here are examples of what I would consider informal chat that would receive little-to-no credit.
Hey Joe, I could not agree with you more! Great job!
Sally, you are so great! I was thinking the same thing.
Sandy, Wow! I never really understood this stuff until I read your post. Thanks!
Tom, I am planning to go to Starbucks later. Would you like to join me?
Joe, I need help! Can you call me at xxx-xxx-xxxx? I need the answers to HW1 so call me on Sunday so that I can submit this homework on time.
Mike, I am so bored. What did you take last semester?
I hope that you can see that these types of replies to other student posts have little-to-no value. They require little thought. Thus, such replies get little-to-no credit. I also do not want to see the following patterns within your replies to other student posts:
Do not restate (or regurgitate) what a student has already stated within his or her main discussion post. Your replies should not come across as just regurgitating that which your classmate has already stated within his or her main discussion post.
Do not pose excessive questions to students when you are replying to their posts. If you have a question and want to post it as a reply, that is fine. Post some preliminary substantive comments and then your question. What I don't want to see are replies to student posts where your reply sounds like an oral exam intended to challenge your classmate.
Do not post excessively long quotes from outside sources. Since this is a discussion forum, I do not want students to copy/paste excessively long quotes from outside sources. Even if you plan to provide in-text citations and corresponding downstream references, your main discussion posts and your replies to other student posts should NOT excessively be composed of quotes taken verbatim from outside sources. I am more interested in hearing your voice within the discussion.
Do not solve the problems of other students. Although I encourage you to show students how to do similar problems within the discussion boards, I do not want you to solve their problems for them.
Do not just post resources. Although I encourage you to provide students with resources, I don't want you to just post a link to a YouTube video or website and treat this as a mathematically substantive reply. If you post a resource, you need to discuss how this resource should be used to help the student. I would recommend providing a concrete math example where you work through the solution and then discuss how a review of your posted resource can help the student solve such math problems.
Provide mathematically substantive replies. Although this is not an exhaustive list, a mathematically substantive reply to another student's post should include one or more of the following items:
Provide a mathematics example where you show your classmate how to perform each step within a particular calculation to illuminate a particular mathematics topic.
Provide a mathematics example where you discuss relevant terminology.
Provide current examples of the credible use of this topic within real-world applications to make informed decisions.
Provide relevant misconceptions (e.g., 23 is sometimes mistakenly solved as 2*3 = 6 instead of 2*2*2 = 8).
Provide misleading ways in which the topic can be interpreted by some students.
Be cordial in your replies to other students. Although it is alright to disagree with your classmates, always be cordial in your replies. In other words, always maintain a warm and friendly tone. The whole point of this discussion board is to create a community of learners where everyone posts in a manner that promotes the learning of mathematics.
I would strongly encourage you to create a draft response for your replies first before posting. Use the spelling, spelling, and grammar checker in Word to make sure that your work is ready to submit.
Discrete Mathematics 3 Response 2
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Discrete Mathematics 3 Response 2
The descriptions of what constitutes proof by contradiction and contrapositive are well-written, although more explanations are required. For instance, the writer failed to indicate if the proof by contradiction is only used for theorems. The contradiction proof is only used for theorems since they can be proven true using axioms and other theorems. We always assume that a proposition is false to eventually identify a contradiction, thereby proving the proposition to be true. We do not use the converse because a contradiction is supposed to imply the truth of the statement. Proof by contradiction involves assuming the proposition is false and showing it leads to a contradiction, not vice versa (Rabin & Quarfoot, 2021). There may be cases when we may assume the proposition is accurate to identify a contradiction, thereby eventually proving that it is false. Proof by contradiction is any argument that verifies a statement through contradiction, even if the supposition is the statement to be proved.
The proof by contrapositive is also missing a few details. For instance, the author failed to describe the relevance of logical equivalence in the proof by contrapositive. Logical equivalence refers to equality between two logical premises or mathematical statements. In mathematics, two logical statements are said to have logical equivalence if they result in the same truth value. Logical equivalence can mean “if and only if verbally”. If and only if statements A and B are logically equivalent, will they return identical truth values? Logical equivalence plays a critical role in proof by contrapositive by providing the basis for inferencing. Proof by contrapositive is a rule of inference that takes advantage of the logical equivalence between “A implies B” and “Not B implies Not A”. We prove “A implies B”...