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By Harris Hancock

Initially released in 1917. This quantity from the Cornell collage Library's print collections used to be scanned on an APT BookScan and switched over to JPG 2000 layout via Kirtas applied sciences. All titles scanned conceal to hide and pages may perhaps comprise marks notations and different marginalia found in the unique quantity.

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10, 000! > 2 · 10 ... , , ... In Chapter VI we derive a formula which can be used to estimate this rapid growth. 14(a). 2 Verify the following equalities using induction: (a) n k=0 k = n(n + 1)/2, n ∈ N. (b) n k=0 k2 = n(n + 1)(2n + 1)/6, n ∈ N. 3 Verify the following inequalities using induction: (a) For all n ≥ 2, we have n + 1 < 2n . (b) If a ∈ N with a ≥ 3, then an > n2 for all n ∈ N. 4 Let A be a set with n elements. Show that P(A) has 2n elements. 44 I Foundations 5 (a) Show that m! (n − m)!

The fiber f −1 (y) is simply the solution set x ∈ X ; f (x) = y of the equation f (x) = y. This could, of course, be empty. 8 Proposition The following hold for the set valued functions induced from f : (i) A ⊆ B ⊆ X = ⇒ f (A) ⊆ f (B). (ii) Aα ⊆ X ∀ α ∈ A = ⇒ f α Aα = α f (Aα ). (iii) (iv) (i ) (ii ) ⇒ f α Aα ⊆ α f (Aα ). Aα ⊆ X ∀ α ∈ A = c A⊆X= ⇒ f (A ) ⊇ f (X)\f (A). A ⊆B ⊆Y = ⇒ f −1 (A ) ⊆ f −1 (B ). Aα ⊆ Y ∀ α ∈ A = ⇒ f −1 α Aα = α f −1 (Aα ). ⇒ f −1 (iii ) Aα ⊆ Y ∀ α ∈ A = α −1 Aα = c α f −1 (Aα ).

Then there is a bijective function from {1, . . , m} to {1, . . , n} if and only if m = n.

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