SECTION 4.1 Basics of Set Theory 197
4.1.4 Mappings between sets
LetAandB be sets. Amappingfrom Ato B is simply a function
fromAto B; we often express this by writingf :A→B or A→f B.
Let’s give some examples (some very familiar):
- f:R→Ris given byf(x) =x^2 −x+ 1, x∈R
- f:R→Cis given byf(x) = (x−1) +ix^2 , x∈R
- LetZ+⊆Zbe the set ofpositiveintegers and defineg:Z^2 →R
byg(m) = cos(2π/n), n∈Z+ - h:R×R→Ris given byf(x,y) =x−y, x, y∈R.
- γ:R×R→Ris given byγ(x,y) =x^2 +y^2
- q:Z→Zis given byq(n) =^12 (n^2 +n), n∈Z
- μ:Z+→{− 1 , 0 , 1 }is given by
μ(n) =
1 ifnis the product of an even number of distinct primes
− 1 ifnis the product of an odd number of distinct primes
0 ifnis not the product of distinct primesThus, for example, μ(1) = 0. Also, μ(6) = 1, as 6 = 2·3, the
product of two distinct primes. Likewise,μ(5) =μ(30) =−1, and
μ(18) = 0.- h:R×R→Cis given byh(x,y) =x+iy, x, y∈R
- σ:{ 1 , 2 , 3 , 4 , 5 , 6 }→{ 1 , 2 , 3 , 4 , 5 , 6 }is represented by
σ:
1 2 3 4 5 6
↓ ↓ ↓ ↓ ↓ ↓
2 5 3 4 1 6