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“Time-Space Tradeoffs for Branching Programs Contrasted with Those for StraightLine Programs,”

A subset R of the Cartesian product of sets is called a relation. A binary relation R is a
subset of the Cartesian product of two sets. Three examples of binary relations are R0 =
{(0, 0),(1, 1),(2, 4),(3, 9),(4, 16)}, R1 = {(red, 0), (green, 1), (blue, 2)}, and R2 =
{(small, short), (medium, middle), (medium, average), (large, tall)}. The relation R0 is a
function because for each first component of a pair there is a unique second component. R1
is also a function, but R2 is not a function.
A binary relation R over a set A is a subset of A × A; that is, both components of each
pair are drawn from the same set. We use two notations to denote membership of a pair (a, b)
in a binary relation R over A, namely (a, b) ∈ R and the new notation aRb. Often it is more
convenient to say aRb than to say (a, b) ∈ R.
10 Chapter 1 The Role of Theory in Computer Science Models of Computation
A binary relation R is reflexive if for all a ∈ A, aRa. It is symmetric if for all a, b ∈ A,
aRb if and only if bRa. It is transitive if for all a, b, c ∈ A, if aRb and bRc, then aRc.
A binary relation R is an equivalence relation if it is reflexive, symmetric, and transitive.
For example, the pairs (a, b), a, b ∈ , for which both a and b have the same remainder on
division by 3, is an equivalence relation. (See Problem 1.3.)
If R is an equivalence relation and aRb, then a and b are said to be equivalent elements.
We let E[a] be the set of elements in A that are equivalent to a under the relation R and
call it the equivalence class of elements equivalent to a. It is not difficult to show that for all
a, b ∈ A, E[a] and E[b] are either equal or disjoint. (See Problem 1.4.) Thus, the equivalence
classes of an equivalence relation over a set A partition the elements of A into disjoint sets.
For example, the partition {0∗, 0(0∗10∗)+, 1(0 + 1)∗} of the set (0 + 1)∗ of binary strings
defines an equivalence relation R. The equivalence classes consist of strings containing zero or
more 0’s, strings starting with 0 and containing at least one 1, an