Mathematical Induction: Find P(sub2)(A(subn)) & Prove (n*(n-1))/2

In summary, the conversation discusses the concepts of sets and subsets, and how the notation P(subk)B represents the set of all subsets of B with exactly k elements. The example given is P(sub2)({1,2,3})={{1,2},{1,3},{2,3}}. The question then asks to find P(sub2)(A(sub1)), P(sub2)(A(sub2)), P(sub2)(A(sub4)), and P(sub2)(A(sub5)). The second part of the conversation introduces the concept of mathematical induction and asks to prove that the number of elements in P(sub2)(A(subn)) is (n*(n-1))/2 for all n elements of Z+.
  • #1
ssome help
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Let A(subn) = {1,2,3,...,n} For any set B, let P(subk)B=the set of all subsets of B with exactly k elements. For example, P(sub2)({1,2,3})={{1,2},{1,3},{2,3}}.

A) Find P(sub2)(A(sub1)), P(sub2)(A(sub2)), P(sub2)(A(sub4)), and P(sub2)(A(sub5))

B) Use mathematical induction to prove that the number of elements in P(sub2)(A(subn)) is (n*(n-1))/2 for all n elements of Z+.
 
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  • #2
Re: Mathematic Induction

ssome help said:
A) Find P(sub2)(A(sub1)), P(sub2)(A(sub2)), P(sub2)(A(sub4)), and P(sub2)(A(sub5))
The goal of this part of the problem is to make sure you understand the notation and what is being asked. Do you understand it? We would rather not give out a complete solution, and to work together we need to know what you understand. If you have questions about what the problem is asking, then say precisely what is not clear.

Edit: In plain text, it is customary to write A_n for $A_n$. If the subscript consists of more than one character, then you need to surround it by parentheses.
 

FAQ: Mathematical Induction: Find P(sub2)(A(subn)) & Prove (n*(n-1))/2

What is the purpose of mathematical induction?

Mathematical induction is a proof technique used to prove that a statement holds for all positive integers. It is based on the principle that if we can prove that the statement holds for the first integer, and then show that if it holds for any arbitrary integer, it must also hold for the next integer, then we can conclude that it holds for all integers.

How is mathematical induction used to find P2(An)?

To find P2(An), we first prove that P2(A1) is true. Then, we assume that P2(Ak) is true for some arbitrary integer k. Using this assumption, we then prove that P2(Ak+1) is true. This shows that if P2(Ak) is true, then P2(Ak+1) must also be true, and by the principle of mathematical induction, we can conclude that P2(An) is true for all positive integers n.

How do you prove (n*(n-1))/2 using mathematical induction?

To prove (n*(n-1))/2 using mathematical induction, we first prove that the statement holds for n=1, which gives us 0 as the result. Then, we assume that the statement holds for some arbitrary integer k. Using this assumption, we prove that the statement also holds for k+1. This involves substituting k+1 in place of n in the original equation and simplifying to show that the result is equal to (k+1)*k/2. This completes the proof by showing that if the statement holds for k, then it must also hold for k+1, and therefore it holds for all positive integers n.

Can mathematical induction be used to prove statements for negative integers?

No, mathematical induction can only be used to prove statements for positive integers. This is because the principle of mathematical induction relies on the fact that if a statement is true for one integer, it must also be true for the next integer. This relationship does not hold for negative integers.

How is mathematical induction related to recursion?

Mathematical induction and recursion are closely related concepts. In mathematical induction, we use the principle of induction to prove that a statement holds for all positive integers by showing that if it holds for one integer, it must also hold for the next integer. This is similar to how recursion works, where a function is defined in terms of itself, and the output for one value is used to calculate the output for the next value. In fact, mathematical induction can be seen as a recursive process of proving a statement for all positive integers.

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