By Terence Tao
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Extra info for Analysis I (Volume 1)
If a set is not equal to the empty set, we call it non-empty. 6 (Single choice). Let A be a non-empty set. Then there exists an object x such that x EA. Proof. We prove by contradiction. Suppose there does not exist any object x such that x E A. Then for all objects x, we have x ¢A. 2 we have x ¢0. 4, a contradiction. 7. 1. Fundamentals 41 demonstrates this non-emptyness. 12) we will show that given any finite number of non-empty sets, say Ab ... , An, it is possible to choose one element x1, ...
A corollary is a quick consequence of a proposition or theorem that was proven recently. 2. Addition 29 m)++· But by definition of addition, 0 + (m++) = m++ and 0 + m = m, so both sides are equal to m++ and are thus equal to each other. Now we assume inductively that n+(m++) = (n+m)++; wenowhavetoshowthat (n++)+(m++) = ((n++)+m)++. The left-hand side is (n + (m++ ))++ by definition of addition, which is equal to ((n+m)++ )++by the inductive hypothesis. Similarly, we have (n++ )+m = (n+m)++ by the definition of addition, and so the right-hand side is also equal to ((n+m)++)++.
Proof. Suppose for sake of contradiction that 6 = 2. 4 we have 5 = 1, so that 4++ = 0++. 4 again we then have 4 = 0, which contradicts our previous proposition. D As one can see from this proposition, it now looks like we can keep all of the natural numbers distinct from each other. 2) allow us to confirm that 0, 1, 2, 3, ... 9. 5, ... }. 4 are still satisfied for this set. 5. But it is difficult to quantify what we mean by "can be obtained from" without already using the natural numbers, which we are trying to define.
Analysis I (Volume 1) by Terence Tao