By Glynn Winskel
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Additional resources for Set Theory for Computer Science [lecture notes]
Boolean propositions are ubiquitous in science and everyday life. They are an unavoidable ingredient of almost all precise discourse, and of course of mathematics and computer science. They most often stand for simple assertions we might make about the world, once we have fixed the meaning of the basic propositional variables. For example, we might take the propositional variables to mean basic propositions such as “It’s raining”, “It’s sunny”, “Dave wears sunglasses”, “Lucy carries an umbrella”, .
Let x, x ∈ X and suppose f (x) = f (x ). Then x = g(f (x)) = g(f (x )) = x . Hence f is injective. Let y ∈ Y . Then f (g(y)) = y. Hence f is surjective. It follows that f is bijective. “only if ”: Assume f : X → Y is bijective. Define the relation g ⊆ Y × X by g = f −1 , the converse relation of f , so (y, x) ∈ g ⇐⇒ f (x) = y. Suppose (y, x), (y, x ) ∈ g. Then, f (x) = y and f (x ) = y, so x = x as f is injective. 2. RELATIONS AND FUNCTIONS 49 making (y, x) ∈ g. This shows that g is a function g : Y → X which moreover satisfies g(y) = x ⇐⇒ f (x) = y .
Dn2 .. dn3 .. ··· .. dni .. ··· .. The decimal expansion of the real r which plays a key role in Cantor’s argument is defined by running down the diagonal of the array changing 1’s to 2’s and non1’s to 1’s. In this way the decimal expansion can never be in the enumeration; no matter which row one considers, the decimal expansion of r will differ on the diagonal. Notice that Cantor’s theorem establishes the existence of irrational numbers, in fact shows that the set of irrational numbers is uncountable, without exhibiting a single irrational number explicitly.