1- Let P(x) denote the statement that log(x3)8 (where x is an integer). Determine whether the following statements are True or False. a) P(2) b) xP(x) c) xxP(x) d) xP(x) e) xP(x) 2- Translate these statements into English, where P(x) is " x is a professor" and C(x) is " x is a clown" and the domain consists of all people. a) x(P(x)C(x)) b) x(P(x)C(x)) c) x(P(x)C(x)) 3- Suppose x is an element from the set {2,3,4} and y is an element from the set {5,6}. Consider the following functions: P(x):2x8 4- Write each of the following propositions using disjunctions, conjunctions, and negations. Then evaluate the truth value of each proposition. a) xP(x) b) x(P(x)) c) xyQ(x,y) d) xyQ(x,y) 5- Solve this inequality 34x+5 and write it using interval notations for x. 6- Prove the following equation logb(x/y)=logb(x)logb(y). Assume b>0,b!=1,x>0,y>0. 7- Prove, by using logical equivalences, that ((acc)(bc))((cb)(ca)) is a tautology. (Do not forget to write the name of the rules that you apply.) 8- Let U be {7,4,2,3} for both x and y. Define P(x,y):x2y>1. What is the truth value of the following? a) xyP(x,y) b) xyP(x,y) c) xyP(x,y) d) xyP(x,y) 9- In the questions below P(x,y) means " x+2y=xy ", where x and y are integers. Determine the truth value of the statement. a) P (1,1) b) P(0,0) c) yy(3,y) d) xyP(x,y) e) xyP(x,y) f) yxP(x,y) g) yxP(x,y) h) xyP(x,y) 10- Express the following statement in predicate logic: "the absolute value of the sum of any two integers is larger than or equal to zero". 11- Express the following statement in predicate logic: "the product of two negative integers is positive