# ProblemSet1

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Homework (1) Giuseppe Aprile Borriello September 15, 2023 1 Exercise 1 ( a ) is a valid conclusion but ( b ) is not a valid conclusion. , if a triangle was isosceles, two sides should be equal but we know that the triangle is not isosceles. all sides are different as mentioned in the definition, scalene triangles have sides of different lengths. ( b ) is not a valid conclusion , the theorem adresses the right triangles, not isosceles or scalene, but the given statement requires the contrary, which is false. 1
2 Exercise 2 1. For all integers x , if x is a prime greater than 2, then x is odd. 2. For all integers x , if x is a set of three consecutive odd integers greater than 3, then not all of the integers in x can be prime. 3. For all integers x , if x is divisible by 8, then x is divisible by 4. 4. For all individuals x , if x stops writing, then x fails. 5. For all people x , if x generally accepts facts as truth, then the facts agree with what x already believes. 2
3 Exercise 3 Claim : x 2 > 0 , x ̸ = 0 , xϵ R Proof : if x > 0, x 2 > 0 if x < 0, ( x ) ( x ) = x 2 > 0 x 2 > 0 3
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