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Mathematical Reasoning and Proof

   

Added on  2020-03-16

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Running head: MATHEMATICAL REASONINGMathematical ReasoningStudentInstitutional Affiliation
Mathematical Reasoning and Proof_1

2MATHEMATICAL REASONINGQUESTION 1With regards to the usual arithmetic operations in conjuction with the property of induction onnutural numbers, addition by reccursion from successor fuction is defined asn+0=n∀n∈Nn+(k+1)=(n+k)+1Our task is to prove that n+k=k+n by inducction on nIt clearly follows that 0+k=k=k+0 since the set of natural numbers is the closure of the emptyset under succesor.NB: The empty set is 0 that is 0=φwhich has no predecessorNow suppose n>0be the successor of ksuch that n=k+1Then it follows that1+n=1+(k+1)¿(1+k)+1 by definition of addition property¿n+1 ................*{Remember 1+n=S(0)+n=S(0+n)=S(n) }Now with assuption that n+k=k+n for some k in the set of natural numbers (k∈N), it canclearly be deduced thatn+(k+1)=(n+k)+1 by definition of additive operation(k+n)+1with regards to the assumption above¿k+(n+1)bydefinitionof+¿ ..................**Hence the proof
Mathematical Reasoning and Proof_2

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