Advanced High-School Mathematics

SECTION 4.1 Basics of Set Theory 203

[4] ={...,− 10 ,− 3 , 4 , 11 , 14 , ...}

[5] ={...,− 9 ,− 2 , 5 , 7 , 12 , ...}

[6] ={...,− 8 ,− 1 , 6 , 7 , 13 , ...}

(ii) Let Rbe the relation on R given by xRy=⇒ x−y ∈Q. This

is easily shown to be an equivalence relation, as follows. First

xRx asx−x = 0 ∈ Q. Next, if xRy, then x−y ∈ Q and so

y−x=−(x−y) ∈Q, i.e.,yRx. Finally, assume thatxRy and

thatyRz. Thenx−y, y−z∈Qand sox−z= (x−y)+(y−z)∈Q

and soxRz. Note that the equivalence class containing the real

numberxis{x+r|r∈Q}.

(iii) Define the functionf :R^2 →Rby settingf(x,y) =x−y. Define

an equivalence relation onR^2 by stipulating that (x 1 ,y 1 )R(x 2 ,y 2 )⇔

f(x 1 ,y 1 ) = f(x 2 ,y 2 ). Note that this is the same as saying that

x 1 −y 1 =x 2 −y 2. Thus, the equivalence classes are nothing more

than the fibres of the mappingf. We can visualize these equiva-

lence classes by noting that the above condition can be expressed

as

y 2 −y 1

x 2 −x 1

= 1, which says that the equivalence classes are pre-

cisely the various lines of slope 1 in the Cartesian planeR^2.

One final definition is appropriate here. LetSbe a set and letRbe
an equivalence relation onS. Thequotient setofSbyRis the set of
equivalence classes inS. In symbols, this is

S/R = {[a]|a∈S}.

We shall conclude this subsection with a particularly important quo-
tient set. Let n ∈ Z+ and let R be the relation “≡ (modn).” One
usually writesZnfor the corresponding quotient set. That is,

Zn = {[m]|m∈Z} = {[0],[1],[2], ...,[n−1]}.^4

(^4) It’s important that the IB examination authors do not use the brackets in writing the elements
ofZn; they simply writeZn = { 0 , 1 , 2 , ..., n− 1 }. While logically incorrect, this really shouldn’t
cause too much confusion.