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
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
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.
When m and n are both even natural numbers.
When m or n 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.
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 = ???