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4. Let n ∈ N. Prove the following identities without using mathematical induction: n (a) ⇓ x − y = (x − y) 3 n xn−j y j−1 . n j=1 n (b) xn + y n = (x + y) (−1)j−1 xn−j y j−1 if n is odd. j=1 n (c) x−n − y −n = (y − x) xj−n−1 y −j if x = 0 and y = 0. S Define 0! = 1 and, for n ∈ N, define n! = n(n − 1) · · · 2 · 1 (n factorial). Prove the following: n! (a) (1 − 1/n)(1 − 2/n) · · · 1 − (n − 1)/n = n . n (2n)! (b) 1 · 3 · 5 · · · (2n − 1) = n . 2 n! 6. ⇓4 For n ∈ Z+ and k = 0, 1, . . , n, define the binomial coefficient n k = n!

If A ⊆ R, we define A+ = {x ∈ A : x ≥ 0}. ♦ Note that by the trichotomy property, a ≤ b and b ≤ a ⇒ a = b. 1) The inequality a ≤ b is sometimes called weak inequality in contrast to strict inequality a < b. The reader may check that parts (a)–(d) of the above proposition are valid if strict inequality is replaced by weak inequality. 4 Definition. The absolute value of a real number x is defined by |x| = x −x if x ≥ 0, if x < 0. ♦ For example, |0| = 0 and |2| = | − 2| = 2. 5 Proposition. Absolute value has the following properties: (a) |x| ≥ 0.

The Real Number System 27 6. S Prove that x 2 = y for · ∞ or · 1 ? 2 1 ≤b x ∞, = (x + y)/2 8. Show that in R3 , a · b = a between a and b. and 2 x ∞ ≤ c x 2. = 1 ⇒ x = y. Is the same true b cos θ, where θ is the (smaller) angle 9. The cross product of vectors a = (a1 , a2 , a3 ) and b = (b1 , b2 , b3 ) in R3 is defined by a2 b2 a×b= a3 a ,− 1 b3 b1 a3 a1 , b3 b1 a2 b2 = a2 b3 − a3 b2 , a3 b1 − a1 b3 , a1 b2 − a2 b1 . Let θ be the (smaller) angle between a and b. Verify the following: (a) (a × b) · a = (a × b) · b = 0.

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