The property is obvious for n = 1. Now assume that the property is valid for n =
ID: 3178770 • Letter: T
Question
The property is obvious for n = 1.
Now assume that the property is valid for n = k
So, in each box with k balls, all balls have equal size. (*)
Now take a box with k+1 balls.
On an arbitrary ball we write the letter A and on another randomly chosen ball we write the letter B. To show that all balls have equal size it is sufficient to prove that the randomly selected balls A and B are equal in size.
Now we pick a ball, different from A and B, from the box. Now, the box contains k balls.
Relying on (*) we know that the k balls are equal in size. So, the balls A and B are equal in size.
Conclusion: In each box with n balls, all the balls are equal in size.
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Explain exactly why this reasoning is wrong. .
We will show that, in each box with n balls, all balls have equal size!Explanation / Answer
Line 1: "The property is obvious for n = 1" Is a True and Acceptable Proposition.
Line 2: "Now assume that the property is valid for n = k
So, in each box with k balls, all balls have equal size. (*)
Is an Assumption of a box having n balls alll are having equal size"
Line 3: Now take a box with k+1 balls.
On an arbitrary ball we write the letter A and on another randomly chosen ball we write the letter B. To show that all balls have equal size it is sufficient to prove that the randomly selected balls A and B are equal in size.
Taking a box with k + 1 balls may contain
(a) all balls have equal size
(b) all balls have different sizes
(c) If the statement is taken from the assumption stated above, i.e., a box with k + 1 balls may contain k identical balls and one odd ball or same size ball (which is clearly not given and cannot be assumed randomly).
Now we pick a ball, different from A and B, from the box. Now, the box contains k balls.
Relying on (*) we know that the k balls are equal in size. So, the balls A and B are equal in size.
Cannot be confirmed that a randomgly selected box of k balls (any box containing k balls ), all balls are having same size. For instance, a box may contain Volley Ball, Food Ball, Base Ball, Table Tennis Ball, etc.. and all are having different sizes and weights.
Conclusion: In each box with n balls, all the balls are equal in size.
Therefore, it may be concluded that, this is a hypothetical question may be true for some boxes having n balls (Existential Quantifier) not for all boxes having n balls (Universal Quantifier)
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