Express each of these statements using predicates, quantifiers, logical connectives, and mathematical operators where the domain consists of all integers.a) The product of two negative integers is positive.b) The average of two positive integers is positive.c) The difference of two negative integers is not necessarily negative.d) The absolute value of the sum of two integers does not exceed the sum of the absolute values of these integers. The contra-positive of $p \rightarrow q$ is $\lnot q \rightarrow \lnot p$. Exercise 51 asks for a proof of this fact.Express the quantification $\exists 1 \times P(x),$ introduced in Section $1.4,$ using universal quantifications, existential quantifications, and logical operators. Greek philosopher, Aristotle, was the pioneer of logical reasoning. Use quantifiers to express each of these statements.a) Everybody can fool Fred.b) Evelyn can fool everybody.c) Everybody can fool somebody.d) There is no one who can fool everybody.e) Everyone can be fooled by somebody.f ) No one can fool both Fred and Jerry.g) Nancy can fool exactly two people.h) There is exactly one person whom everybody can fool.i) No one can fool himself or herself.j) There is someone who can fool exactly one person besides himself or herself. The truth tables of each statement have the same truth values. A proposition is a collection of declarative statements that has either a truth value "true” or a truth value "false". The converse of $p \rightarrow q$ is $q \rightarrow p$. Let $Q(x, y)$ be the statement "x has sent an e-mail message to $y,$ " where the domain for both $x$ and $y$ consists of all students in your class. It has two parts −. collection of declarative statements that has either a truth value \"true” or a truth value \"false [Hint: Use logical equivalence from Tables 6 and 7 in Section $1.3,$ Table 2 in Section $1.4,$ Example 19 in Section $1.4,$ Exercises 47 and 48 in Section $1.4,$ and Exercises 48 and $49 . PROPOSITION (OR) STATEMENT: Proposition is a declarative statement that is either true or false but not both. (The new variable $y$ is used to combine the quantifications correctly.). Propositional Logic is concerned with statements to which the truth values, “true” and “false”, can be assigned. Next, express the negation in simple English. Thus the inverse of $p \rightarrow q$ is $ \lnot p \rightarrow \lnot q$. Express the negations of these propositions using quantifiers, and in English.a) Every student in this class likes mathematics.b) There is a student in this class who has never seen a computer.c) There is a student in this class who has taken every mathematics course offered at this school.d) There is a student in this class who has been in at least one room of every building on campus. If the statement is “If p, then q”, the inverse will be “If not p, then not q”. The connectives connect the propositional variables. Express each of these statements using mathematical and logical operators, predicates, and quantifiers, where the domain consists of all integers.a) The sum of two negative integers is negative.b) The difference of two positive integers is not necessarily positive.c) The sum of the squares of two integers is greater than or equal to the square of their sum.d) The absolute value of the product of two integers is the product of their absolute values. Express each of these quantifications in English.$$\begin{array}{ll}{\text { a) } \exists x \exists y Q(x, y)} & {\text { b) } \exists x \forall y Q(x, y)} \\ {\text { c) } \forall x \exists y Q(x, y)} & {\text { d) } \exists y \forall x Q(x, y)} \\ {\text { e) } \forall y \exists x Q(x, y)} & {\text { f) } \forall x \forall y Q(x, y)}\end{array}$$, Let $P(x, y)$ be the statement "Student $x$ has taken class $y,$ where the domain for $x$ consists of all students in your class and for $y$ consists of all computer science courses at your school.