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Blast into Math! ets of nummers: mathematical plaagrounds
Definition 3.2.1. The natural numbers are
1, 2, 3, 4, 5,...
Starting from 1, the next natural number is 2= 1+ 1. For each natural number n there is a next natural
number which is equal to n +1. The set of natural numbers is
N = {1, 2, 3, 4,...,n,n +1,...}.
The set N of natural numbers is unending because starting from 1, there is always a next natural number.
Definition 3.2.2 Let S be a set. If for any n ∈ N , there are at least n elements in S , then we define the
number of elements in S to be infinite; equivalently, we say that S has infinitely many elements and write
#S = ∞.
Infinity (which can be written as ∞ ) is then defined to be the number of elements in an infinite set.
Proposition3.2.3 The set of natural numbers has infinitely many elements.
Proof: For each n ∈ N there are n natural numbers, from 1 up to n . Therefore, for each n ∈ N there
are at least n elements in the set N . So, by the definition, the number of elements in N is infinite.
We can think of the set of natural numbers as our big, infinite playground! What games may we play
within the safety of this playground? We can add or multiply natural numbers, and the result is still
a natural number. This is like playing the games of addition and multiplication without leaving the
playground. More precisely, let’s define what it means for a set to be closed under a binary operation.
Definition 3.2.4 A binary operation on a set is an operation that we can do with two elements of the set.
A set is closed under a binary operation if, when we perform the operation on two elements of the set, the
result we get from the operation is also an element of the set.
Proposition 3.2.5 The natural numbers are closed under the binary operations addition and multiplication.
th
Proof: Let x and y be natural numbers. Then, x + y is the y next natural number after x , so
x + y ∈ N . xy is x added to itself y times. Each time we add x to itself, the result is a natural
number. So, eventually, the result of adding x to itself y times is also a natural number.
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