Semantics Set Theory Set Theory gave us a way to create expressions that could represent the meaning of words and simple sentences Question 12 (5pt) Suppose we have the set ???? = {1, 3, 5, {7, 8, 9}, 12}. (a) What is the cardinality of ????? (ie, what is |????|?) (b) Give three distinct subsets of ????. (c) Which relation(s) (from , , or ) could you put in the blank to make this expression true? {7, 8, 9} ???? (d) Which relation(s) (from , , or ) could you put in the blank to make this expression true? 3 ???? (e) Which relation(s) (from , , or ) could you put in the blank to make this expression true? {12, 3} ???? Question 13 (2pt) Suppose we have three sets ????, ????, and ????, and we know that ???? (???? ????). What can we say about any set relation that must hold between ???? and ????? Give an example of ????, ????, and ???? that have these relationships. Question 14 (2pts) Suppose we define the following sets: ???? : { , , , , , } ???? : {, , } (a) Give an enumeration of (???? ???? ) (b) Give an enumeration of (???? ???? ) (c) What is the truth value of (???? ???? )? (d) What is the truth value of (???? ????)? Question 15 (2pt) Suppose we define the following representations from Set Theory: R = { Rover } ("Rover" is the name of a dog) D = {x | x is a dog} B = {x | x is brown} G = {x | x is big} Give an English sentence that is represented by the following Set Theoretic expression: (???? (???? (???? ????))) Question 16 (2 pts) For the complex sentence below, create an expression of propositional logic that represents the meaning of the whole sentence, using atomic propositions and meaning connectors. Then use Set Theory to give a representation of the meaning of each atomic proposition. Try to give a Set Theoretic expression for each atomic proposition in your propositional logic expression. If you cannot create a Set Theoretic expression to represent the meaning of any of your atomic propositions, you must (i) explain what the problem is that prevents you from representing that atomic proposition using Set The- ory, and (ii) define the atomic proposition that cannot be represented in Set Theory with a simple declarative sentence of English. Example: Roses used to be red, and violets are not blue. You could represent the meaning of thiscomplex sentence with this expression of propositional logic: (????1 ????2) representing your atomic propositions using Set Theory as: ???? = {x | x is a rose} ???? = {x | x is red} ???? = {x | x is a violet} ???? = {x | x is blue} ????1: the meaning representation for "roses used to be red"; we cannot express the meaning of this sentence with something like (???? ????), because Set Theory cannot represent things that change with time, which is part of the meaning of "used to be" ????2: violets are blue = (???? ????) (a) Whether red apples are expensive or not, you should not buy green apples.