LOGIC AND PROOFS . Express each of these statements in terms of quantifiers and then determine its truthvalue.a) There is a student in the class who is a junior.b) Every student in the class is a computer science major.c) There is a student in the class who is neither a mathematics major nor a junior.d) Every student in the class is either a sophomore or a computer science major.e) There is a major such that there is a student in the class in every year of study with that major. Let L(x, y) be the statement “x loves y,” where the domain for both x and y consists of all people in the world. It has two parts −. If the statement is “If p, then q”, the converse will be “If q, then p”. The connectives connect the propositional variables. It has many practical applications in computer science like design of computing machines, artificial intelligence, definition of data structures for programming languages etc. Use quantifiers to express each of these statements.a) Everybody loves Jerry.b) Everybody loves somebody.c) There is somebody whom everybody loves.d) Nobody loves everybody.e) There is somebody whom Lydia does not love.f ) There is somebody whom no one loves.g) There is exactly one person whom everybody loves.h) There are exactly two people whom Lynn loves.i) Everyone loves himself or herself.j) There is someone who loves no one besides himself or herself. The Discrete Mathematics Notes pdf – DM notes pdf book starts with the topics covering Logic and proof, strong induction,pigeon hole principle, isolated vertex, directed graph, Alebric structers, lattices and boolean algebra, Etc. Use quantifiers to express the distributive laws of multiplication over addition for real numbers. Exercise 51 asks for a proof of this fact.Show how to transform an arbitrary statement to a statement in prenex normal form that is equivalent to the given statement. Use quantifiers to express each of these statements.a) Lois has asked Professor Michaels a question.b) Every student has asked Professor Gross a question.c) Every faculty member has either asked Professor Miller a question or been asked a question by Professor Miller.d) Some student has not asked any faculty member a question.e) There is a faculty member who has never been asked a question by a student.f ) Some student has asked every faculty member a question.g) There is a faculty member who has asked every other faculty member a question.h) Some student has never been asked a question by a faculty member. a) Show that $\forall x P(x) \wedge \exists x Q(x)$ is logically equivalent to $\forall x \exists y(P(x) \wedge Q(y)),$ where all quantifiers have the same nonempty domain.b) Show that $\forall x P(x) \vee \exists x Q(x)$ is equivalent to $\forall x \exists y$ $(P(x) \vee Q(y)),$ where all quantifiers have the same nonempty domain. A proposition is a collection of declarative statements that has either a truth value "true” or a truth value "false". Translate these statements into English, where the domain for each variable consists of all real numbersa) $\exists x \forall y(x y=y)$b) $\forall x \forall y(((x \geq 0) \wedge(y<0)) \rightarrow(x-y>0))$c) $\forall x \forall y \exists z(x=y+z)$. $(A \land B) \lor (A \land C) \lor (B \land C \land D)$, "Man is Mortal", it returns truth value “TRUE”, "12 + 9 = 3 – 2", it returns truth value “FALSE”. A Contingency is a formula which has both some true and some false values for every value of its propositional variables. (Assume that all e-mail messages that were sent are received, which is not the way things often work. For example, $\exists x \forall y(P(x, y) \wedge Q(y))$ is in prenex normal form, whereas $\exists x P(x) \vee \forall x Q(x)$ is not (because the quantifiers do not all occur first).Every statement formed from propositional variables, predicates, $\mathbf{T},$ and $\mathbf{F}$ using logical connectives and quantifiers is equivalent to a statement in prenex normal form. Propositional Logic is concerned with statements to which the truth values, “true” and “false”, can be assigned. The domain in each case consists of all real numbers.a) $\exists x \forall y(x y=y)$b) $\forall x \forall y(((x<0) \wedge(y<0)) \rightarrow(x y>0))$c) $\exists x \exists y\left(\left(x^{2}>y\right) \wedge(x

2020 logic and proofs in discrete mathematics