For each chosen problem in the following sections, identify at least one fallacy and explain in detail why the fallacy leads to a contradiction. You must identify a fallacy, not merely a contradiction which results from a fallacy. You must interpret each problem in good faith as intended without misrepresenting it or dismissing it on an irrelevant technicality. Clearly number each problem.

Choose two of these beginner (B) problems. Find at least one fallacy in each chosen problem (10% each). B1 Let a and b be any two integers such that a = b. B3 Let x be any positive integer. Then P b given X = 1+1+1+ .+1 Q2 = ab multiply by like term x-many addends a2 - b2 = ab - b2 subtract like term x(x) = x(1+1+1+ .+1) (a+ b) (a-b) = b(a-b) factor each side = x+x+x+ .+x a+b = b divide by like term If we let f(x) = x2 then, by calculus, atb-b b - b subtract like term a - 0 simplify f (x) = x2 f(x) = x+x+x+"+x Since a was a completely arbitrary integer, it seems that all f(x)' = 1+1+1+ ."+1 integers are actually equal to zero. f(x)' B2 Observe the following by algebra. However, also by calculus, -1 given f(x) = x2 -1/1 = -1/1 divide by like term f(x)' = 2x -1/1 1/-1 adjust sign placement Then, by substitution on f(x)' above, -1/1 = V1/-1 square root each side 2x = X F-INVI = VI/V-1 simplify 2x - X X- X i/1 1/i substitute constants, simplify X 0 1/i simplify Note that since x was any positive integer, we must conclude 1 multiply by like term that all positive integers are equal to zero. -1 substitute constants II simplify Therefore 1 and -1 are equal