Math Test 1

This week in Math, we had our first test, so I thought I would make a test for you. Since I don’t want to put you off your breakfast, if the thought of a Math test makes you break out in Hives, then feel free to skip this week’s post. For the rest of you, I have Questions followed by example Answers. I’ve tried to make sure my answers are accurate, but if you find a problem, feel free to link:mailto:setupminimal@gmail.com[let me know].

The Test

What is a Statement?

A statement is a sentence which is either true or false. For example: the previous sentence.

What is an Open Sentence?

An open sentence is a sentence, along with a domain, which becomes a statement when each item in the domain is placed into the sentence. For example: For x in {1, 2, 3, 7}, x*2 - 10 < 0.

Is the statement “If the Sun is purple then today is Tuesday” true or false?

True! When dealing with an implication, if your condition is incorrect information (The Sun isn't purple), then it doesn't matter what the rest of the sentence is, the whole condition is true. My professor helps us remember this by saying "Well, it isn't the _logic_ that's flawed."

Proove that when x and y are integers, x is odd or y is even if and only if (x – 3)(y + 4) is even.


To prove the above, we must show that:

  1. if x is odd or y is even, (x – 3)(y + 4) is even.
  2. if (x – 3)(y + 4) is even, x is odd or y is even.

To prove 1, we will use a direct proof by cases.

Case A: Assume y is even. Then y + 4 is even, so (x – 3)(y + 4) is even. Case B: Assume x is odd. Then x – 3 is even, so (x – 3)(y + 4) is even.

To prove 2, we will use a contrapositive proof. Assume that x is even and y is odd. Then x = 2k for some integer k, and y = 2j + 1 for some integer j. Therefore, (x – 3)(y + 4) = (2k – 3)(2j + 1 + 4) = (2k – 3)(2j + 5). Since niether of those factors is even, the whole expression is odd. Therefore if (x – 3)*(y + 4) is even, then either x is odd or y is even.

Since 1 and 2, it is prooven.

Comments

Our actual test was longer, but I have no interest in boring you with a full recounting. The important thing is to remember to occasionally check your knowledge to make sure that you haven’t developed any gaps of understanding. The professor has emphasized that from here on proof writing becomes a larger part of the course. I hope to include more proofs like the above, both as examples and to help me write clearer proofs through practice.

I hope to see you all again next week for more mathematics!