Calculus for mathematicians, computer scientists, and by Hwang A.

By Hwang A.

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Example text

Velocity is supposed to represent “instantaneous rate of change of position,” but what exactly does this mean? ) The ancient Greek Zeno of Elea discovered the following apparent paradox. ” The stone has a definite location, and is indistinguishable from a stationary stone at the same height. More concretely, imagine an infinitely fast camera that captures literal instants of time. Then there is no way to distinguish a moving object from a stationary object on the basis of a single photograph. But since this argument can be made at every instant of time, there is no difference between moving and standing still!

The subset ∆ is often called the diagonal of X × X. 11 Let X be a set having more than one element. 7, namely by R = (X ×X)\∆ = {(x, y) ∈ X ×X : x = y}. 12 Let X be a set. An equivalence relation on X is a relation, usually denoted ∼, such that • (Reflexivity) For all x ∈ X, x ∼ x. In words, every element is related to itself. • (Symmetry) For all x and y ∈ X, x ∼ y if and only if y ∼ x. Roughly, the relation sees only whether x and y are related or not, and does not otherwise distinguish pairs of elements.

2 says A(ℓ) is true for all ℓ ∈ N, that is, addition is associative. Commutativity is proven by a double application of induction; first show that n + 1 = 1 + n for all n ∈ N (by induction on n), then prove that n + m = m + n for all m, n ∈ N (by induction on m). Associativity is used several times. Consider the statement P (n) : n + 1 = 1 + n. The base case P (1) says the successor of 1 is the successor of 1 (or 1 + 1 = 1 + 1), which is obviously true. Now assume P (k) is true for some k ∈ N; we wish to prove P (k + 1).

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