Section 4.1 Sets
229
Definition BiNaRy OPeRatiON
+ is a binary operation on a set S if for every ordered pair (x, y) of elements of S,
x + y exists, is unique, and is a member of S.
In other words, if + is a binary operation on S, then for any two values x and
y in S, x + y produces one and only one answer, and that answer belongs to S.
That the value x + y always exists and is unique. It is described by saying that
the binary operation + is well-defined. The property that x + y always belongs to
S is described by saying that S is closed under the operation +. Uniqueness does
not mean that the result of a binary operation occurs only once; it means that for
a given x and y, there is only one result. For subtraction, there are many x and y
values such that x − y = 7, but for a given x and y, like x = 5 and y = 2, there is
only one answer for x − y.
example 7
Addition, subtraction, and multiplication are all binary operations on ℤ. For
example, when we perform addition on the ordered pair of integers (x, y), x + y
exists and is a unique integer.
example 8
The logical operations of conjunction, disjunction, implication, and equivalence
are binary operations on the set of propositional wffs. If P and Q are propositional
wffs, then P ` Q, P ~ Q, P S Q, and P 4 Q are unique propositional wffs.
A candidate + for an operation can fail to be a binary operation on a set S in
any of three ways: (1) There are elements x, y [ S for which x + y does not exist; (2)
there are elements x, y [ S for which x + y gives more than one result; or (3) there
are elements x, y [ S for which x + y does not belong to S.
example 9
Division is not a binary operation on ℤ because x ÷ 0 does not exist.
example 10
Define x + y on ℕ by
x + y = e
1 if x ≥ 5
0 if x ≤ 5
Then, by the first part of the definition for +, 5 + 1 = 1, but by its second part,
5 + 1 = 0. Thus, + is not well-defined on ℕ because the result of 5 + 1 is not
unique.
example 11
Subtraction is not a binary operation on ℕ because ℕ is not closed under subtraction. (For example, 1 − 10 o ℕ.)
229
Definition BiNaRy OPeRatiON
+ is a binary operation on a set S if for every ordered pair (x, y) of elements of S,
x + y exists, is unique, and is a member of S.
In other words, if + is a binary operation on S, then for any two values x and
y in S, x + y produces one and only one answer, and that answer belongs to S.
That the value x + y always exists and is unique. It is described by saying that
the binary operation + is well-defined. The property that x + y always belongs to
S is described by saying that S is closed under the operation +. Uniqueness does
not mean that the result of a binary operation occurs only once; it means that for
a given x and y, there is only one result. For subtraction, there are many x and y
values such that x − y = 7, but for a given x and y, like x = 5 and y = 2, there is
only one answer for x − y.
example 7
Addition, subtraction, and multiplication are all binary operations on ℤ. For
example, when we perform addition on the ordered pair of integers (x, y), x + y
exists and is a unique integer.
example 8
The logical operations of conjunction, disjunction, implication, and equivalence
are binary operations on the set of propositional wffs. If P and Q are propositional
wffs, then P ` Q, P ~ Q, P S Q, and P 4 Q are unique propositional wffs.
A candidate + for an operation can fail to be a binary operation on a set S in
any of three ways: (1) There are elements x, y [ S for which x + y does not exist; (2)
there are elements x, y [ S for which x + y gives more than one result; or (3) there
are elements x, y [ S for which x + y does not belong to S.
example 9
Division is not a binary operation on ℤ because x ÷ 0 does not exist.
example 10
Define x + y on ℕ by
x + y = e
1 if x ≥ 5
0 if x ≤ 5
Then, by the first part of the definition for +, 5 + 1 = 1, but by its second part,
5 + 1 = 0. Thus, + is not well-defined on ℕ because the result of 5 + 1 is not
unique.
example 11
Subtraction is not a binary operation on ℕ because ℕ is not closed under subtraction. (For example, 1 − 10 o ℕ.)