What in the world does an upside down mean? (Set relation proof)

[Race]

Homework Statement

Let A = (upside down !) (whatever that is). Consider the following rlation R on this set: xRy iff x=y+n for some integer n.

a. Prove R an equivalence relation on A

b. If R is an equiv. relation, find an explicit description of the elements in [1] and [1.25].
Bonus: What are the elements in [3.1415] Explain what elements are in the equivalence class of a generic real number z? (ie: explicitly describe [z]).

The Attempt at a Solution

I know that to be an equivalence relation it must be reflexive, transitive, and symmetric, but I can't even get to that part when I'm stuck on what the upside-down-! is supposed to be. All numbers in existence ever? In a question later on on the study sheet he says upside-down-! represents the set of all real numbers. Am I supposed to apply that to this question too?

And then there is B, which completely confuses me because I had an infection - which is the very reason I'm not doing this sheet in class with classmate group help.

Any help you guys can give me would be extremely appreciated!


Ps. Um... I think I sound bitter. I'm really not. Really. :D

 

[kurt.physics]

A. To prove R is an equivalence relation, we must show that it is reflexive, symmetric, and transitive. Reflexive: For any element x in the set A, xRx. That is, x = x + 0, which is true for all real numbers. Symmetric: If xRy, then yRx. That is, if x = y + n for some integer n, then y = x - n, which is true for all real numbers. Transitive: If xRy and yRz, then xRz. That is, if x = y + n and y = z + m, then x = z + (n + m). This is also true for all real numbers. Therefore, R is an equivalence relation on A. B. The elements in [1] are 1, 1 + n, where n is any integer. The elements in [1.25] are 1.25, 1.25 + n, where n is any integer. Bonus: The elements in [3.1415] are 3.1415, 3.1415 + n, where n is any integer. The elements in the equivalence class of a generic real number z are z, z + n, where n is any integer.