7. Consider the following statements: P(x, y) = "x R such that Vy R, x+y=0" Q(x, y) = "VxER 3y R such that x+y=0" (a) Explain the difference between what these two statements are claiming. Which of them is true and which of them is false?
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- 3 5. Say for each of the following sets of statements if they are consistent or inconsistent. A.G Ↄ H / H Ↄ J / J Ↄ K/ K Ↄ ~G b. P ≡ Q / Q v P / P Ↄ X / Q Ↄ ~P c. Z Ↄ [(P ● R) v (L ● P)] / ~P Ↄ ~Z / L Ↄ~ RMake use of De Morgan’s Laws to write the given statement in an equivalent form. 6. It is not true that Bob failed the English proficiency test and he registered for a speech course.Write the following compound statement in terms of the simple statements p, q, and r. "I go to office hours and ask questions, and my professor does not remember me." p: I go to office hours. q: I ask questions. r: My professor remembers me.
- For each of the following statements, determine whether they are true or false. If false, write their logical negation (distributing the ¬ across any expressions as necessary) and explain how this negation is true. In each of these, x and y are assumed to be integers. (a) (∀x)(∀y)((y > x) ⇒ (x = 0)) (b) (∃x)(∀y)((y > x) ⇒ (x = 0)) (c) (∀x)(∃y)((y > x) ⇒ (x = 0)) (d) (∀x)(((∀y)(y > x)) ⇒ (x = 0))1. Give the unifyiing substitition for (x,y,z are variables): f(a) f(b) 2. Give the unifyiing substitition for (x,y,z are variables): g(x,x) g(f(a),f(y)) 3. Give the unifyiing substitition for (x,y,z are variables): g(x,x) g(f(a0,f(b))1. State the value of x after the staement if P(x) then x := 1 is executed, where P(x) is the statement "x>1," if the value of x when this statement is reached is x=0 2. Let Q(x) be the statement " x+1 > 2x." If the domain consists pf the integers, what are these truth values? ∀xQ(x) 3. Determine the truth value of each of these statements if the domain of each variable consists of all real numbers. ∀x(x^2+2≥1) 4. suppose that the domain of the propositional function P(x) consists of the integers -2,-1,0,1, and 2. Write out the proposition using disjunctions, conjunctions, and negations. ∀xP(x) 5. Translate this statement into a logical expressions using predicates, quantifiers, and lofical connectives. Something is not in the correct place.
- 10. Let p be the statement “He gets frustrated.” and q be the statement “He will be able to complete the job.” Translate ~(p ∧ q) into the equivalent statement.1. a) By using logic laws, determine if the statement below is a contradiction, tautology or neither. [p→(q→p)] ↔ (pᴧ¬p) b) By using logic laws, test if the argument below is valid: Either I exercise regularly, or I eat healthy. If I exercise regularly then I will not get fat. I got fat. Therefore, I am healthy.Do this question: Fill in the blanks so that the argument in part (b) has the same logical form as the argument in part a: a. If n is divisible by 6, then n is divisible by 3. If n is divisible by 3, then the sum of the digits of n is divisible by 3. Therefore, if n is divisible by 6, then the sum of the digits of n is divisible by 3. (b). If this function is ______________ then this function is differentiable. If this function is ______________ then this function is continuous. Therefore, if this function is a polynomial, then this function ______________. a. If all computer programs contain errors, then this program contains an error. This program does not contain an error. Therefore, it is not the case that all computer programs contain errors. b. If ______________, then ______________. Tom is not Python Programmer. Therefore, it is not the case that all software programmers in the company use Python for programming. do it without Ai
- For the following statement, write down a logically equivalent statement which containsno operators other than ¬ and ∨: P ∧ (Q → R)Considering p, r and q are statements, use logic laws to indicate: (pvq) →r ≡ (p→r) ᴧ (q→r) By using logic laws, determine if the statement below is a contradiction, tautology or neither. [p→(q→p)] ↔ (pᴧ¬p) By using logic laws, test if the argument below is valid: Either I exercise regularly, or I eat healthy. If I exercise regularly then I will not get fat. I got fat. Therefore, I am healthy.8. Let ?, ?, and ? be true, false and false, respectively. Determine the truth value of the following? a.) (? → ?) ∧∼ ? b.) ? ↔ (? ∧ ?)