By Terence Tao

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This closes the induction. 0 We now discuss how addition interacts with positivity. 7 (Positive natural numbers). A natural number n is said to be positive iff it is not equal to 0. l). 8. 4). Proof. We use induction ·on b. If b = 0, then a + b = a + 0 = a, which is positive, so this proves the base case. Now suppose inductively that a+ b is positive. 3, and is hence positive. This 0 closes the induction. 9. If a and bare natural numbers such that a+b 0, then a = 0 and b = 0. = 2. 2. Addition 31 Proof.

4444 ... a real number, while ... 321 is not? And why do we have to carry of digits when adding or multiplying? 999 ... the same number as 1? What is the smallest positive real number? 00 ... 001? So to set aside these problems, we will not try to assume any knowledge of the decimal system, though we will of course still refer to numbers by their familiar names such as 1,2,3, etc. instead of using other notation such as 1,11,111 or 0++, (0++)++, ((0++)++)++ (see below) so as not to be needlessly artificial.

If A, B, A' are sets, and A is equal to A', then A U B is equal to A' U B (why? 4). Similarly if B' is a set which is equal to B, then A U B is equal to A U B'. Thus the operation of union obeys the axiom of substitution, and is thus well-defined on sets. We now give some basic properties of unions. 13. If a and b are objects, then {a, b} ={a} U {b}. , (AUB)UC = AU (BUG)). Also, we have A U A = A U 0 = 0 U A = A. 1. 3. 4, we need to show that every element x of (AUB)U C is an element of A U (B U C), and vice versa.

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