Problem 9.4. Study the following claim as well as the alleged \"proof\" Claim. I

Question

Problem 9.4. Study the following claim as well as the alleged "proof" Claim. If A and B are infinite sets, then A B "Proof". Let A and B be infinite sets. Then we can describe A and B as follows A-(a1.2.... an.)and B-(b.b..n Define a function f: A-B by f(an)bn for all an E A. Clearly,fis one-to-one and onto B. Therefore A B, finishing the proof Complete the following questions concerning the above claim and "proof" (1) Determine whether the "proof" is rigorous. Identify the issues in the "proof" (2) Determine whether the claim is true or false. Justify the answer in part (3) (3) If the the claim is true and the "proof" is not rigorous, then provide a correct and rigorous proof. If the claim is false, give a concrete counterexample.

Explanation / Answer

1) suppose A has n elements and n tends to infinity .

and B has n+1 elements and then both the sets will not be having same no. elements

and thus A is not equivalent to B. and neither the function f is onto.

So this is not a rigourous proof .

2) Claim is true.

3) it should be included that infinity + 1 =infinity .

infinity * infinity = infinity , infinity is not any number , its a concept .

thus in part 1 , n and n+1 both are same when n is infinity .

thus the cardinality is same for A and B is same according to this concept . and the similar way f can be made one one and onto . Since cardinality is same , A is equivalent to B.

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