# MAT_2125_HW1Q_V2 (1) (1).pdf - Homework 1 MAT 2125 1...

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Homework 1: MAT 2125 January 18, 2021 1 Practicing L A T E X 1. We could re-write the English sentence “For every real number, there exists a bigger real number” as x R , y R : y > x. Do a similar translation of the English sentence “For every integer, there exists a smaller integer.” 2. Compute R 10 0 sin( x ) cos( x ) dx , showing at least two intermediate steps. Using the “align” environment or otherwise, vertically align the “=” sign between steps. 2 Cardinality Show that there exists a bijection between Z and Q . Hint: If you can find a surjection in both directions, then you have shown that a bijection exists. This might be easier. 3 Calculations with Axioms 1. Using only the field axioms, show that the multiplicative identiy is unique. That is, show that if a, b are both multiplicative identities, then in fact a = b . 2. Using only the field axioms, show that (2 x - 1)(2 x + 1) = 4 x 2 - 1. Note: The field axioms don’t define 2 or 4 are. Please take these to be shorthands for 2 = 1 + 1 and 4 = 1 + 1 + 1 + 1.