Hi, not sure if this is the right category just saw logic and set theory in the info so I assumed that this would be the proper place.

I'm currently taking a class called MAT231 Sets, functions, and relations.

I honestly can't remember when I ever did a proof before (think high school but it was skippable on the exam).

Below is the assigned worksheet I'm suppose to do #1,#2.

#3 and #4 are suppose to be more challenging ones that you can attempt.

I've tried to attempt parts of #1

a. Yes, there is 64 total squares with even dimensions of 8 x 8.

8^2 = 2x

x= 32

b. No, if you remove opposite corners then the dimensions would change. Creating a 6 by 6 inner square with a border of of 7 on each side. The dominoes can fill the 6 by 6 inner square but cannot cover the "odd" dimensions of the border.

Let S = inner square

S = 6^2

S= 36

Lets T = border

T= 7+7+7+7 = 28....Error in my logic this contradicts since 28 is divisible by 2 (length of a domino) but I don't believe the board can be complexly covered because of the change in dimensions.

c. No, the demensions are odd.

d.

When mandn are both even natural numbers.

When morn is an even natural number.

e. When m is an odd natural number and n is an even natural number. (or viseversa)

I really need help this is the first assignment and all we talked about briefly in the first class was "conjectures" which we were told is an educated guess. I lack how to find the logic for the statements I need.

#2

a. 1 = 1; 1 + 3 =4; 1 + 3 + 5 =9; 1 + 3 + 5 + 7 =16;

b. 1 + 3 + ... + (2n + 1) = ???

c. 1 = 1; 1 + 8 =9; 1 + 8 + 27 =36; 1 + 8 + 27 + 64 =100;

d. 1 + 8 + ... + n^3 = ???