5.3.6 Here is a 'proof' that all human beings have the same age. Where is the flaw in the argu- ment? 1) In a set with only 1 person, all the people in the set have the same age. Proof. (Base case n = (Inductive hypothesis) true that all of the people in the set have the same age. Suppose that for some integer n > 1 and for all sets with n people, it is (Inductive step) Let A be a set with n+1 people, say A = {a1,...,an, an+1}, and let A' = {a1,...,an} and A" = {a2, ...,an+1}. The inductive hypothesis tells us that all the people in A' have the same age and all the people in A" have the same age. Since a2 belongs to both sets, then all the people in A have the same age as a2. We conclude that all the people in A have the same age. (Conclusion) By induction, the claim holds for all n > 1.

5.3.6 Here is a 'proof' that all human beings have the same age. Where is the flaw in the argu- ment? 1) In a set with only 1 person, all the people in the set have the same age. Proof. (Base case n = (Inductive hypothesis) true that all of the people in the set have the same age. Suppose that for some integer n > 1 and for all sets with n people, it is (Inductive step) Let A be a set with n+1 people, say A = {a1,...,an, an+1}, and let A' = {a1,...,an} and A" = {a2, ...,an+1}. The inductive hypothesis tells us that all the people in A' have the same age and all the people in A" have the same age. Since a2 belongs to both sets, then all the people in A have the same age as a2. We conclude that all the people in A have the same age. (Conclusion) By induction, the claim holds for all n > 1.

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Description: 5.3.6 Here is a 'proof' that all human beings have the same age. Where is the flaw in the argu-ment?Proof. (Base case n = 1) In a set with only 1 person, all the people in the set have the same age.(Inductive hypothesis) Suppose that for some intege

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Permutations and Combinations

If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.

Counting Theory

The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.

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5.3.6 Here is a 'proof' that all human beings have the same age. Where is the flaw in the argu-
ment?
Proof. (Base case n = 1) In a set with only 1 person, all the people in the set have the same age.
(Inductive hypothesis) Suppose that for some integer n > 1 and for all sets with n people, it is
true that all of the people in the set have the same age.
(Inductive step) Let A be a set with n +1 people, say
A = {a1,...,an, an+1}, and let
A' = {a1,... , an} and A" = {a2,. , an+1}.
The inductive hypothesis tells us that all the people in A' have the same age and all the people
in A" have the same age. Since a2 belongs to both sets, then all the people in A have the same
age as a2. We conclude that all the people in A have the same age.
(Conclusion) By induction, the claim holds for all n > 1.

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