Hey all, I have a test next week after Spring Break and would like some extra help with these problems in order to make sure I am prepared for the actual test!
1. Use a proof by cases to show that for all real numbers x and y, the absolute value of X+Y is less than or equal to the absolute value of X + the absolute value of Y.
For this I am thinking simply breaking into cases using X is + or -, and Y is + or -. That seems fairly simple. If X is - and Y is - the absolute value of X+Y is equal to the absolute value of X + the absolute value of X. The same applies for X and Y both being +. If one is + and the other -, the absolute value of X + Y is going to be less than the absolute value of X + the absolute value of Y. Am I missing anything here? Once I actually write out my proof I'll be sure to post it to make sure it makes sense.
2. Using problem 1, prove that for all real numbers A and B, the absolute value of A-B is greater than or equal to the absolute value of A - the absolute value of B.
Is this not essentially the exact same thing as 1?
3. Use a contradiction argument to show that for all positive real numbers X,Y, and Z, if X>Z and Y^2=XZ, then X>Y>Z. Hint: X>Y>Z is equivalent to saying X>Y and Y>Z.
Contradiction arguments use the general form of p and not q implies r and not r, correct? So for this problem it would be if X>Z and Y^2=XZ and X less than or equal to Y less than or equal to Z. Is this correct?
4. Prove that if D is an odd integer that divides both A+B and A-B, then D divides both A and B.
The definition of divides is that there A|B if there is some integer K we can multiply A by to get B, correct? So for this problem it would be if K_1D=A+B and K_2D=A-B then K_3D=A and K_4D=B correct? (The underscores represent subscripts).
5.Prove that N is odd if and only if N^2 is odd.
The definition of odd is that a number N can be represented by 2K_1+1 yes? So for an if and only if argument we prove if N is odd then N^2 is odd, then we prove the converse of if N^2 is odd, N is odd?
6. Prove that the sum of two odd numbers is even.
So I simply add (2K_1 +1)+(2K_2+1) and simply from there?
7. Prove by way on contradiction that for integers A,B, and C, if A^2+B^2=C^2, then at least one of A and B is even. Hint, use problem 5 and 6.
This seems pretty overwhelming but I'm sure it will be easier once I have a better grasp of 5 and 6.
8. Prove that for all real numbers X and Y, if Y^3+YX^2 less than or equal to X^3+XY^2, then Y less than or equal to X. Hint, use a proof by contrapositive.
Contrapositive proofs reverse and negate the implication right? So for this it would be if Y>X, then Y^3+YX^2>X^3+XY^2?
9. Prove that the square root of 3 is irrational.
A rational number can be expressed as A/B where A and B are both integers. I'm not sure how to start this as I don't know how to express an irrational number.
10. 7^n -2^n is divisible by 5 for all natural numbers n. Hint, add a clever form of zero when doing the inductive step.
I can set the base case up for this fairly easily but I have no idea what the clever form of zero is during the inductive step.
11. (1+1/2)^n greater than or equal to 1+n/2.
This is another induction problem but I don't know how to deal with the inequality. Are there any particular tips I should use for these that don't apply to equalities?
12.Suppose x+1/x is an integer. Prove that x^2+1/x^2,...x^n+1/x^n are all integers.
Another induction problem. Once I really get into it I imagine I'll come back with some questions but nothing seems terribly awful about it just looking at it.
13. Let P and Q be positive integers with Q>P. Prove that Q-P divides Q-1 if and only if Q-P divides P-1.
I don't even know what process to use on this one. I know the if and only if means I will be using the converse. But where do I even start?
14. A student claims to have shown that 2=1. Critique the student's proof by finding the error in their argument. Explain your reasoning below.
Proof: Let a=b be an integer. Then we have the following.
Things start to look strange around step 4/5. However, I'm not sure what is wrong. Obviously 2 doesn't equal 1...but I don't see the error.
Thanks for anyone willing to help me out, I would like to do well on the test