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Note that if we don't have the Extremal Clause, 0.5, 1.5, 2.5, ... can be included in N, which is not what we want as the set of natural numbers.
Example 2. Definition of the Set of Nonnegative Even Numbers NE
The set NE is the set that satisfies the following three clauses:
Basis Clause: 0 ∈ NE
Inductive Clause: For any element x in NE, x + 2 is in NE.
Extremal Clause: Nothing is in NE unless it is obtained from the Basis and Inductive Clauses.
Example 3. Definition of the Set of Even Integers EI
The set EI is the set that satisfies the following three clauses:
Basis Clause: 0 ∈ EI
Inductive Clause: For any element x in EI, x + 2, and x - 2 are in EI.
Extremal Clause: Nothing is in EI unless it is obtained from the Basis and Inductive Clauses.
Example 4. Definition of the Set of Strings S over the alphabet {a,b} excepting empty string. This is the set of strings consisting of a's and b's such as abbab, bbabaa, etc.
The set S is the set that satisfies the following three clauses:
Basis Clause: a ∈ S, and b ∈ S.
Inductive Clause: For any element x in S, ax ∈ S, and bx ∈ S.
Here ax means the concatenation of a with x.
Extremal Clause: Nothing is in S unless it is obtained from the Basis and Inductive Clauses.
Tips for recursively defining a set:
For the "Basis Clause", try simplest elements in the set such as smallest numbers (0, or 1), simplest expressions, or shortest strings. Then see how other elements can be obtained from them, and generalize that generation process for the "Inductive Clause".
The set of propositions (propositional forms) can also be defined recursively.
As we saw earlier, union, intersection and Cartesian product of sets are associative. For example (A ∪ B) ∪ C = A ∪ (B ∪ C)
To denote either of these we often use A ∪ B ∪ C.
This can be generalized for the union of any finite number of sets as A1 ∪ A2 ∪.... ∪ An.
which we write as
${}_{i=1}^{n}{A}_{i}$
This generalized union of sets can be rigorously defined as follows:
Definition ( ${}_{i=1}^{n}{A}_{i}$ ):
Basis Clause: For n = 1, ${}_{i=1}^{n}{A}_{i}={A}_{1}$ .
Inductive Clause: ${}_{i=1}^{n+1}{A}_{i}$ = ${}_{i=1}^{n}{A}_{i}$ ∪ An+1
Similarly the generalized intersection ${}_{i=1}^{n}{A}_{i}$ and generalized Cartesian product ${}_{i=1}^{n}{A}_{i}$ can be defined.
Based on these definitions, De Morgan's law on set union and intersection can also be generalized as follows:
Theorem (Generalized De Morgan)
$\overline{{}_{i=1}^{n}{A}_{i}}={}_{i=1}^{n}\overline{{A}_{i}}$ , and
$\overline{{}_{i=1}^{n}{A}_{i}}={}_{i=1}^{n}\overline{{A}_{i}}$
Proof: These can be proven by induction on n and are left as an exercise.
Some functions can also be defined recursively.
Condition: The domain of the function you wish to define recursively must be a set defined recursively.
How to define function recursively: First the values of the function for the basis elements of the domain are specified. Then the value of the function at an element, say x, of the domain is defined using its value at the parent(s) of the element x.
A few examples are given below.
They are all on functions from integer to integer except the last one.
Example 5: The function f(n) = n! for natural numbers n can be defined recursively as follows:
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