Practice Questions on Recursive Definition

## Recursive Definition

### Subjects to be Learned

• recursive/inductive definition
• basis clause
• basis
• inductive clause
• extremal clause

### Contents

Sets which have too many elements to list them up, and for which there is no convenient or obvious predicates to specify their elements can often be defined using a recursive definition (also called inductive definition). It essentially gives a procedure to generate the members of the set one by one starting with some subset of its elements.
In this type of definition, first a collection of elements to be included initially in the set is specified. These elements can be viewed as the seeds of the set being defined. Next, the rules to be used to generate elements of the set from elements already known to be in the set are given. These rules provide a method to construct the set element by element starting with the seeds. These rules can also be used to test elements for the membership in the set.

A recursive definition of a set always consists of three distinct clauses:
1. The basis clause (or simply basis) of the definition establishes that certain objects are in the set. This part of the definition specifies the "seeds" of the set from which the elements of the set are generated using the methods given in the inductive clause. The set of elements specified here is called basis of the set being defined.
2. The inductive clause (or simply induction) of the definition establishes the ways in which elements of the set can be combined to produce new elements of the set. The inductive clause always asserts that if objects are elements of the set, then they can be combined in certain specified ways to create other objects. Let us call the objects used to create a new object the parents of the new object, and the new object is their child .
3. The extremal clause asserts that unless an object can be shown to be a member of the set by applying the basis and inductive clauses a finite number of times, the object is not a member of the set.

### Examples of Recursive Definition of Set

Example 1. The Set of Natural Numbers

Basis Clause:
Inductive Clause: For any element x in , x + 1 is in .
Extremal Clause: Nothing is in unless it is obtained from the Basis and Inductive Clauses.

The basis for this set N is { 0 } . The x + 1 in the Inductive Clause is the parent of x, and x is the child of x + 1.
Following this definition, the set of natural numbers N can be obtained as follows:
First by (1),   0 is put into N.
Then by (2), since 0 is in N,  0 + 1 (= 1) is in N. 0 is the parent of 1, and 1 is the child of 0.
Then by (2) again,   1 + 1 (= 2) is in N. 1 is the parent of 2, and 2 is the child of 1.
Proceeding in this manner all the natural numbers are put into N.
Note that if we don't have (3),  0.5, 1.5, 2.5, ... can be included in N, which is not what we want as the set of natural numbers.

Here we are assuming without stating it that the number 0 exists and the operation of addition on numbers can be used. However, we do not like to make too many assumptions. So far we have made only one assumption. That is, sets exist. The set of natural numbers can be defined without making any further assumptions. The concept of numbers and addition are captured using sets and set operations. Click here to see how it is defined.

Example 2. The Set of Nonnegative Even Numbers

Basis Clause:
Inductive Clause: For any element x in , x + 2 is in .
Extremal Clause: Nothing is in unless it is obtained from the Basis and Inductive Clauses.

Example 3. The Set of Even Integers

Basis Clause:
Inductive Clause: For any element x in , x + 2, and x - 2 are in .
Extremal Clause: Nothing is in unless it is obtained from the Basis and Inductive Clauses.

Example 4. The Set of Strings over the alphabet excepting empty string
This is the set of strings consisting of a's and b's such as abbab, bbabaa, etc.

Basis Clause: , and .
Inductive Clause: For any element x in , , and .
Here ax means the concatenation of a with x.
Extremal Clause: Nothing is in 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. To see how it is defined click here.

### Test Your Understanding of Recursive Definition

Indicate which of the following statements are correct and which are not.
Click Yes or No , then Submit. There are two sets of questions.

Next -- Generalized Set Operations

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## Recursive Definition of Function

### Subjects to be Learned

• recursive definition of function

### Contents

Some functions can also be defined recursively.

Condition: The domain of the function you wish to define recursively must be 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:

Basis Clause: f(0) = 0! = 1
Inductive Clause: For all natural number n,  f(n+1) = (n+1) f(n).

Note that here Extremal Clause is not necessary, because the set of natural numbers can be defined recursively and that has the extremal clause in it. So there is no chance of other elements to come into the function being defined.

Using this definition, 3! can be found as follows:
Since 0 ! = 1,   1 ! = 1 * 0 ! = 1 * 1 = 1 ,
Hence 2 ! = 2 * 1 ! = 2 * 1 = 2 .
Hence 3 ! = 3 * 2 ! = 3 * 2 * 1 = 6 .

Example 6: The function f(n) = 2n + 1 for natural numbers n can be defined recursively as follows:

Basis Clause: f(0) = 1
Inductive Clause: For all natural number n,  f(n+1) = f(n) + 2 .
See above for the extremal clause.

Example 7: The function f(n) = 2n for natural numbers n can be defined recursively as follows:

Basis Clause: f(0) = 1
Inductive Clause: For all natural number n,  f(n+1) = 2 f(n) .
See Example 5 for the extremal clause.

Example 8: The function L from the set S of strings over {a, b} to the set of natural numbers that gives the length of a string can be defined recursively as follows:

Basis Clause: For symbols a and b of the alphabet,   L(a) = 1 and L(b) = 1.
Inductive Clause: For any string x and y of S,  L(xy) = L(x) + L(y) ,  where xy is the concatenation of strings x and y.
See Example 5 for the extremal clause.

This function L gives the number of a's and b's

Next -- Recursive Algorithm

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