Section 1: (Binary) Operations
It is assumed that you learned in Math 250 about sets and mathematical induction, the content
of Section 0, and we will begin with binary operations. Technically:
Def: For a set S, any function S×S→Sis a (binary) operation. (Such creatures as “unary
operations” exist, i.e., functions S→S, as well as “ternary”, S×S×S→S, etc., but we are only
interested in the binary ones, so we will usually drop the “binary” part of the term.)
So we know many, many (binary) operations on IR; for example, (a, b)7→ a3+ sin(3b+ 7). But
of course the term is intended to capture under a single word some familiar ideas that have some
things in common while still not being identical:
•Addition, multiplication and subtraction of real numbers are all operations on IR. (Division
is not, because division by 0 isn’t defined, so division is only a function IR ×(IR − {0})→IR);
•Addition of n-vectors and cross-product of 3-vectors are operations on IRnand IR3. (Dot
product is not, because it is a function IRn×IRn→IR.)
•Addition of matrices of the same dimensions, and multiplication of square matrices of the
same dimension are operations on Mm×n(IR) and Mn×n(IR) respectively.
•“Pointwise” addition and multiplication of real-valued functions on some common set (e.g.,
an interval in the real line) is an operation on the set of such functions: The function f+g
is defined on the common domain by (f+g)(x) = f(x) + g(x) for every xin the domain.
We might talk about these in calculus I, discussing definite integrals: For integrable functions
f, g on the interval [a, b], Rb
a(f+g)(x)dx == Rb
af(x)dx +Rb
ag(x)dx.
•For any positive integer n, two operations on the set ZZn={0,1,2, . . . , n −2, n−1}of possible
remainders on long division by n, called “integers modulo n” are addition and multiplication
modulo n: For a, b in ZZn,a⊕band abare the remainders when the integers a+band a·b
are computed in the usual way in ZZ and then divided by n. (It is assumed that you learned
about ZZnin Math 250.)
•Compositions of functions from a set Xto itself is an operation: If f , g are functions X→X,
then f◦gis defined on Xby (f◦g)(x) = f(g(x)) for every xin X;
•On a finite set, an operation can be defined by a table: On a set S={a, b, c, d},
∗a b c d
a a b c d
b d c b a
c c c d d
d d d c c
means
a∗a=a a ∗b=b a ∗c=c a ∗d=d
b∗a=d b ∗b=c b ∗c=b b ∗d=a
c∗a=c c ∗b=c c ∗c=d c ∗d=d
d∗a=d d ∗b=d d ∗c=c d ∗d=c
•For a fixed set X, the family S=P(X) of all subsets of Xhave at least three natural
operations on it. (The script P is because Sis called the “power set” of X.): For subsets
A, B of X,
A∪B={x∈S:x∈Aor x∈Bor both}
A∩B={x∈S:x∈Aand x∈B}
A4B={x∈S:x∈Aor x∈Bbut not both}= (A∪B)−(A∩B)
