To prove: n∈N∧2n+1∈N⟹∃m∈N:n=2m+1. (natural numbers, implication, existence, odd number)
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Step-by-Step Solution
Step 1: Understand the Given Information
We are given two conditions: n is a natural number, and 2n+1 is also a natural number. Natural numbers are typically positive integers (1,2,3,…). Some definitions include 0 as a natural number, but for this proof, positive integers are sufficient.
Step 2: Define a New Variable
Since 2n+1 is a natural number, we can represent it with a variable, say k. Setting k=2n+1 implies that k must be a natural number because the given statement tells us it is.
Step 3: Rearrange the Equation
Now we rearrange the equation k=2n+1 to express n in terms of k. First, multiply both sides by 2 to get 2k=n+1. Then, subtract 1 from both sides to isolate n, resulting in n=2k−1. This form is crucial for proving the final statement.
Step 4: Examine the Properties of k
Since k is a natural number, its smallest possible value is 1 (assuming N={1,2,3,…}). This means k is a positive integer.
Step 5: Manipulate the Expression for n
We want to show that n can be written in the form 2m+1 for some natural number m. We can rewrite 2k−1 as 2(k−1)+1. This form directly relates to the target expression for n.
Step 6: Define m and Verify it is a Natural Number
Let m=k−1. Since k is a natural number and k≥1, then m=k−1 will be greater than or equal to 0. If k=1, then m=0. Some definitions of natural numbers include 0, while others do not. However, the standard definition of an odd number requires m to be an integer (non-negative). If n=1, then 1=2(0)+1, so m=0. If the definition of natural numbers for m includes 0, then m∈N. If it strictly means positive integers, this requires clarification. For example, in many contexts, 0 is included in the set of non-negative integers or whole numbers, and n=2m+1 naturally leads to m=0 for n=1. For the purpose of this proof, we will assume that the definition of natural numbers for m allows for 0 (i.e., N∪{0} or Z≥0). If m=0 is not considered a natural number, then the statement would require n>1 for m to be a 'natural' number.
Step 7: Conclude the Proof
By substituting m=k−1 into the expression for n, we get n=2m+1. Since k is a natural number, m=k−1 is a non-negative integer. If the definition of natural numbers includes zero, then m∈N. If not, then m∈Z≥0. In either case, the form n=2m+1 correctly defines n as an odd number, which is what the implication means.