dis6b.pdf - CS 70 Fall 2019 Discrete Mathematics and Probability Theory Alistair Sinclair and Yun S Song Quiz 6 1[True or False(a The set of all

dis6b.pdf - CS 70 Fall 2019 Discrete Mathematics and...

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CS 70 Discrete Mathematics and Probability Theory Fall 2019 Alistair Sinclair and Yun S. Song Quiz 6 1. [True or False] (a) The set of all irrational numbers R \ Q (i.e. real numbers that are not rational) is uncountable. (b) The set of integers x that solve the equation 3 x 2 ( mod 10 ) is countably infinite. (c) The set of real solutions for the equation x + y = 1 is countable. For any two functions f : Y Z and g : X Y , let their composition f g : X Z be given by f g = f ( g ( x )) for all x X . Determine if the following statements are true or false. (d) f and g are injective (one-to-one) = f g is injective (one-to-one). (e) f is surjective (onto) = f g is surjective (onto). 2. Consider an n × n matrix A where the diagonal consists of alternating 1’s and 0’s starting from 1, i.e. A [ 0 , 0 ] = 1 , A [ 1 , 1 ] = 0 , A [ 2 , 2 ] = 1 , etc. Describe an n length vector from { 0 , 1 } n that is not equal to any row in the matrix A . (Note that the all ones vector or the all zeros vector of length n could each be rows in the matrix.) 3. Find the precise error in the following proof: False Claim: The set of rationals
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