EXERCISE |1.6.2: Truth values for quantified statements about integers. ? In this problem, the domain is the set of all integers. Which statements are true? If an existential statement is true, give an example. If a universal statement is false, give a counterexample. (a) ax (x + x= 1) (b) 3x (x + 2.= 1) ( L # X - ZX) XA (5) (0# X - ZX) XA (P) (e) vx ( x2 > 0) (f) =x (x2 > 0) Feedback? EXERCISE 1.6.3: Translating mathematical statements in English into logical expressions. ? Consider the following statements in English. Write a logical expression with the same meaning. The domain is the set of all real numbers. (a) There is a number whose cube is equal to 2. (b) The square of every number is at least 0. (c) There is a number that is equal to its square. (d) Every number is less than or equal to its square plus 1. Feedback?Feedback? EXERCISE 1.6.4: Truth values for quantified statements for a given set of predicates. (? The domain for this problem is a set {a, b, c, d}. The table below shows the value of three predicates for each of the elements in the domain. For example, Q(b) is false because the truth value in row b, column Q is F. P Q R T T F T F F C T F T F F Which statements are true? Justify your answer. (x)dXA (e) (b) 3x P(x) (x)0 XA (0) (X)O XE (P) (X) XA (2) (f) 3x R(x) Feedback? EXERCISE 1.6.5: Converting a quantified expression to an equivalent logical expression. (?(e) vx R(x) (f) ax R(x) Feedback? EXERCISE 1.6.5: Converting a quantified expression to an equivalent logical expression. P(x) is a predicate and the domain for the variable x is {1, 2, 3, 4). For each of the logical expressions given, give an equivalent logical expression that does not use quantifiers. (x)d XA (e) (b) ax P(x) Feedback? How was this section? 6 1 Provide feedback ivity summary for assignment: Participation and Challenge Activities 1.6 & 1.7 42 / 42 pt 07/30/2022, 12:00 AM PDT 42 / 42 pts submitted to canva pletion details v MacBook Prolog Help/FAC Check Next Feedback? Additional exercises EXERCISE 1.7.1: Determining whether a quantified statement about the integers is true. ? Predicates P and Q are defined below. The domain is the set of all positive integers. P(x): x is prime Q(x): x is a perfect square (i.e., x = y2, for some integer y) Indicate whether each logical expression is a proposition. If the expression is a proposition, then give its truth value. (a) ax Q(x) ( x) d- V (x)OXA (9) (E)d A (x)OXA (0) (d) 3x (Q(x) / P(x) ) ((x) d / ( x ) 0 - ) XA (2 ) Feedback? (?) EXERCISE 1.7.2: Translating quantified statements from English to logic, part 1. In the following question, the domain is a set of students at a university. Define the following predicates:Gm My library > COMP 047: Discrete Math for Computer Science home > 1.7: Quantified statements zyBooks catalog Help EXERCISE 1.7.4: Translating quantified statements from English to logic, part 3. In the following question, the domain is a set of employees who work at a company. Ingrid is one of the employees at the company. Define the following predicates: S(x): x was sick yesterday W(x): x went to work yesterday V(x): x was on vacation yesterday Translate the following English statements into a logical expression with the same meaning (a) At least one person was sick yesterday. (b) Everyone was well and went to work yesterday. (c) Everyone who was sick yesterday did not go to work. (d) Yesterday someone was sick and went to work. (e) Everyone who did not go to work yesterday was sick. (f) Everyone who missed work was sick or on vacation (or both). (g) Someone who missed work was neither sick nor on vacation. (h) Each person missed work only if they were sick or on vacation (or both). (i) Ingrid was sick yesterday but she went to work anyway. () Someone other than Ingrid was sick yesterday. (Note for this question, you will need the expression (x # Ingrid).) (k) Everyone besides Ingrid was sick yesterday. (Note that the statement does not indicate whether or not Ingrid herself was sick yesterday. Also, for this question, you will need the expression (x * Ingrid).)