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Greetings MTH 225 students. This is an online space for you to ask anything about proof, probability, or anything else in the class. You can **search** for keywords in the upper right corner; use Markdown syntax for text highlighting; and use LaTeX for math notation. I'll leave this room open for your questions and comments for the rest of the semester.

$\binom{52}{5}$

The following are some overall comments and suggestions about HWB Proofs, for your consideration as you do any revisions you need to do. Not all problems are represented here; just ones that had widespread attempts.

**General remarks:**

*Before writing anything in a proof, outline it.*Choose the method of proof you want to use; clearly state the assumptions; clearly state what it is you are going to prove having made the assumptions.*Be very careful not to assume what you are trying to prove.*This results in an automatic "Repeat" mark because circular reasoning is not a proof.*Do not give specific examples in place of a proof*; and do not include specific examples in a proof (exception: the base case of an induction proof).- When proving that an equation or inequality is true, do not start off by writing down the equation and inequality and then "work backwards". This is assuming what you are trying to prove. This is incorrect because we don't know that the starting point is true yet. Instead
*start with just one side of the proposed equation or inequality and do work to arrive at the other side*.

**Problem 1:** General comment -- if you are taking square roots, it will end poorly. This proof can and should be done without any root-taking whatsoever. Instead think about the definition of "divisibility". Also recall that we proved in class that if $x^2$ is even, then $x$ is even.

**Problem 3:** Most of the errors on this problem would have been avoided by being clearer at the outset about what proof technique is being used, what the assumptions are, and what is going to be proven. If you have to revise that one, this would be an important first step.

**Problems 5--8 generally:** Please note the fourth bullet above -- do not prove an equation or inequality by starting with the equation or inequality. Instead *start with just one side of the proposed equation or inequality and do work to arrive at the other side . Also __clearly state the induction hypothesis* when it's time to do so. Without a clearly stated induction hypothesis the proof is incorrect.

**Problems 5 and 6:** Beware that when you proceed to the induction step that you write down exactly the right statement to be proven. "We will now prove this for k+1" is not precise enough to be correct. Also it involves more than just adding 1 to everything.

**Problem 8:** There are two variables in the expression but only one of them is used by the predicate, namely $n$. The variable $x$ is not used in the induction process.

Now, use the chat feature to ask questions -- this won't be threaded and so you may have to scroll to see other people's questions and responses, but we'll try to maintain order anyway.