MTH105 Discrete Mathematics

Question 1(a)Let 𝑝𝑝  and𝑞𝑞  be  statements.  Employ  a  truth  table  to  verify that  the  proposition  q)(~∧∨pp is a tautology. b)Use the rules of inference to show the validity of the following argument form. 𝑝𝑝⟶(~𝑞𝑞)(premise)𝑟𝑟⟶𝑞𝑞(premise)𝑟𝑟(premise)∴~𝑝𝑝(conclusion)Do not use truth tables in this part of the question and give a reason for each step.Question 2(a)Consider the following argument:All student must take a writing course. Carol is a student.  Therefore, Carol must take a writing course.Let  S(x) and  W(x)  be  the  predicates  “x  is  a  student”  and  “x  takes  a  writing  course”,  respectively.(i) Rewrite the above argument using the predicates S(x) and W(x) together with the universal and/or existential quantifier.  (ii)State whether the argument is valid or invalid and give the name to justify your choice.b)Consider the following statement P : For all real numbers x and y, if 22yx=thenyx=. (i) Write statement  P  using  universal  and/or  existential  quantifier,  together  with any necessary logical connectives.(ii)Also write the negation of statement P using universal and/or existential quantifier