Both parts of question 1 in this attached document.

 

CS-2102Discrete MathProblem Set 4, Due Thursday February 25, 2016.Remember that you can work in groups that are no larger than 4.Be sure to give adequateexplanation for your solutions, lack of clarity and completeness may result in points beingdeducted.Answers given with no work or justification will have points taken off.(Hint: for the next 2 problems look at the end of the set of slides on Predicates.)(4 pts, 2 pts each) 1. Rewrite each of the following statements in the two forms “∀°,if____________ then ____________” and “∀_____________°, _____________” (without the if-then).(a)All equilateral triangles are isosceles.(b)Every computer science student needs to take data structures.(4 pts, 2 pts each) 2. Rewrite the following statements in the two forms “∃____________°such that __________” and “∃°such that ___________ and ____________.”(a)Some hatters are mad.(b)Some questions are easy.(18 pts, 3 each) 3. Determine if the following statements are true or false.Prove the statementdirectly from the definition(s) if true, and give a counterexample if it is false.(a)For all integers?,?, and?, if?divides?then?divides??.(b)4 divides the product of any two even integers.(Note the definition of even is: aninteger±is even if and only if∃² ∈ ℤsuch that± = 2².)(c)A sufficient condition for an integer to be divisible by 8 is that it be divisible by 16.(d)For all integers?,?, and?, if?|?and?|?then?|(2? − 3?).(e)For all integers?,?, and?, if??|?then?|?and?|?.(f)For all integers?,?, and?, if?|(? + ?)then?|?or a|c.(4 pts) 4.Find a counterexample to the statement that every positive integer can be written asthe sum of the squares of three integers.