Differences Between Inductive Set and Inductive Proof

An Inductive Set may be described as a set in which every element is related to the other element as every element in the inductive Set has a predecessor and a successor. Now we also saw the definition of The Inductive Proof. An Inductive Proof may be described as a mathematical proof in which if the function holds true for a particular element, then it must also hold true for the next element in the sequence. Now we will look further into these concepts.

Inductive Set

As we have already seen earlier that an Inductive Set is one in which every element is related to each other, every element in the Inductive Set has a predecessor and a successor provided that they are not the first and last elements. An example of The Inductive Set is the set of Natural Numbers. This is because, in the set of Natural Numbers, every element has a successor and a predecessor. That is why the set of Natural Numbers can be considered an Inductive Set. Similarly, if we were to take another example of Inductive Sets, we could consider the example of The Set of Roll Numbers of students in a particular class. This is because, in general, the roll numbers of students are in serial order and each element in the set of Roll Numbers will be related to another element because the roll numbers are in order. 

Inductive Sets have some constraints. That is, in order for a set to be an Inductive Set, the set should be nonempty. This means that the set should have one or more elements. The second constraint is that the set must be partially ordered. In Mathematics, a partially ordered set is a set whose elements are in some kind of order or the set whose elements follow some kind of rules related to their position in the set. The third and the most important constraint for a set to be an Inductive Set is that in the set every element should have a successor, this basically means that the elements should be related to each other. One more way of looking at it is, that successive elements are related to each other through some kind of rule.

Inductive Proofs

Inductive Proof is a proof in which if the proof holds true for a particular element, it also holds true for the next element in the sequence. This basically means that, if the proof is true for an element a, then it should also be true for the element a+1.

Let’s consider a function P, and let’s consider an element a

If P (a) is true, then P (a+1) should be true in order for function P to be an Inductive Function.

Difference between Inductive Sets and Inductive Proofs

There is a major difference between Inductive Sets and Inductive Proofs. Inductive Set is a mathematical set whereas Inductive Proof is a mathematical proof. In Mathematics, a set can be defined as a group of objects which are similar in type to each other. An example of a set can be the set of natural numbers or the set of heights of students in a particular class. Whereas a mathematical proof is a mathematical statement that is used to prove equations. Now we have seen a major difference between Inductive Sets and Inductive Proofs. We have come to know that they are both different Mathematical Quantities. An inductive Set is a set whereas Inductive Proof is proof. Inductive Set is a particular Set in which each element is related to the other element in that one element is a predecessor of the other element and one element is the successor of the other element. Whereas Inductive Proof is a Mathematical Proof that if true for one particular element, has to be true for the next element in the sequence. So these are the few differences between Inductive Sets and Inductive Proofs. Now we will see some similarities between these two concepts.

Although Inductive Sets and Inductive Proofs are as different as day and night, they both do operate on the concept and that concept is the concept of Induction. An inductive Set is a set whereas Inductive Proof is proof. Inductive Set is a particular Set in which each element is related to the other element in that one element is a predecessor of the other element and one element is the successor of the other element. Whereas Inductive Proof is a Mathematical Proof that if true for one particular element, has to be true for the next element in the sequence. From these statements, we can clearly see that Inductive Sets and Inductive Proofs use the same concept of Induction.

Conclusion

In this article, we have defined both Inductive Sets and Inductive Proofs. We have also seen how both the concepts of Inductive Sets and Inductive Proofs are in detail. Then we took a look at the differences as well as the similarities between the two topics of Inductive Sets and Inductive Proofs. We have seen in the discussion that both inductive set and inductive proof use the concept of induction.