Q2: You try it! Select any one TRUE and any one FALSE true/ false question from any of the practice exams that use IMPLIES (aka an if-then statement). For each of the two problems, identify the propositions in the problem (the "if part" and the "then part"). If the implication is false, provide a counter-example with explanation/ calculations to show why it works. If the implication is true, give a short general* explanation using precise and correct terminology from class. (* a short general proof - not an example) Note: please write the problem statement of the problem you are solving, to help the grader. Finally, some statements can have one or more mathematical quantifiers. There are two kinds of mathematical quantifiers, which are the universal quantifier for all (aka for every), and the existential quantifier for some (aka there exists). For example, each of the following statements has a quantifier. (quantifier highlighted)10. If AT = b is consistent for some b E Rm, then A has a pivot in every row. 11. If A does not have a pivot in every column and T(@) = Ax, then for every 1 and X2 with x1 + 2 we have that T(1) = T(T2).Q3: You try it! In (10.) and (11.) identify the propositions in each problem. The implications (10.) and (11.) are both false; provide a counterexample to one of the implications. Give a short description (one or two short sentences) as to why your counterexample works, and verify any assertions you make with calculations. Note: please write the problem statement of the problem, you are solving, to help the grader. Hint: For (10.) a counterexample will be a matrix A and a vector 5 such that A5? = h is consistent, but A does not have a pivot in every row. Hint: For (11.) a counterexample will be a matrix A that does not have a pivot in every column, and such that there exist vectors fl and 551 such that fl # :32 and T(:E'1) 7E T(:'2)