1. Prove Rule 11: A+ÃB = A+B. starting from the left side A+ĀB and prove it's equal to the right side A+B 2. Simplify the following Boolean expression: a. ĀBC + AB C + Ā B C+ ABC + ABC Ans: BC + AB + B C b. (Ā+B) (A+B+D) D Ans: BD
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A: Identity Identity Complement Complement Idempotent Idempotent Annulment
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Q: State h1 h2 3 3 A 3 2 1 3 4 4 G
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- Translate each of these nested quantifications into an English statement that expresses a mathematicalfact. The domain in each case consists of all real numbers.a) ∃x∀y(x + y = y) b) ∀x∀y (((x ≥ 0) ∧ (y < 0)) → (x − y > 0))c) ∃x∃y(((x ≤ 0) ∧ (y ≤ 0)) ∧ (x − y > 0))d) ∀x∀y((x ≠ 0) ∧ (y ≠ 0) ↔ (xy ≠ 0))Determine whether the following is true or false. Please cite the brief explanation so that I can know why is it true or false. A) 1 ∈ {{1},{2},{3}} B) A\B = A ∩ B^c C) The contrapositive of p -> (p V ¬p) = (¬p∧ q) -> ¬pWrite the First Order Logic of the following:(Hint: you may need to use these symbols: Vx, Ix, A, 7, V)(a) Tim and Mike like to play FIFA, Mike beats Tim.(b) All Muslims go to Mosques.(c) Some kids like to go to Disney Land and other kids likes to go to Lego Land.(d) Kuala Lumpur is the capital city of Malaysia.(e) Only one student missed the quiz and the final exam.(f) All the students in ITIB3914 class are intelligent.(g) Mike likes Al class and does not like math class.(h) John is the father of Mike.(i) Kate is the mother of Sara.(i) John and Kate are parents.
- Translate these statements into English, where C(x) is “x is a comedian” and F(x) is “x is funny” andthe domain consists of all people.a) ∀x(C(x) → F(x))b) ∀x(C(x) ∧ F(x))c) ∃x(C(x) → F(x))Correct answer will be upvoted else downvoted. Computer science. in the event that |a|>0 (the length of the string an is more prominent than nothing), erase the principal character of the string a, that is, supplant a with a2a3… an; in the event that |a|>0, erase the last character of the string a, that is, supplant a with a1a2… an−1; if |b|>0 (the length of the string b is more prominent than nothing), erase the main character of the string b, that is, supplant b with b2b3… bn; in the event that |b|>0, erase the last character of the string b, that is, supplant b with b1b2… bn−1. Note that after every one of the tasks, the string an or b might become vacant. For instance, in the event that a="hello" and b="icpc", you can apply the accompanying grouping of tasks: erase the principal character of the string a ⇒ a="ello" and b="icpc"; erase the primary character of the string b ⇒ a="ello" and b="cpc"; erase the principal character of the string b ⇒…Simplify the following assertions (so that ¬ does not appear). a) ¬((∃a ∈ A,((∀b ∈ B, a × b = 2) ⇒ (∃b ∈ B, a + b ≠ 3))) & (∀a ∈ A, ∃b ∈ B, ((a+b = 5)∨(a−b = 5)))) b) ¬((∃a ∈ A, ∀b ∈ B, ((a + b = 3) ∨ (a - b = 3))) & (∀a ∈ A, ((∃b ∈ B, a × b = 7) ⇒ (∀b ∈ B, a + b ≠ 4)))) c) ¬((∃a ∈ A, ∀b ∈ B, ((a + b = 2) & (a - b = 2))) ∨ (∀a ∈ A, ((∃b ∈ B, a + b = 7) ⇒ (∀b ∈ B, a + b = 5)))) d) ¬((∀a ∈ A,((∃b ∈ B, a - b = 4) ∨ (∀b ∈ B, a + b ≠ 2))) & (∃a ∈ A, ∀b ∈ B, ((a+b = 5)∨(a−b = 5))))
- Solve the following puzzle by translating statements into logical expressions and reasoning fromthese expressions.Four friends have been identified as suspects for an unauthorized access into a computer system.They have made statements to the investigating authorities.1-Alice said Carlos did it.2-John said I did not do it.3-Carlos said Diana did it.4-Diana said Carlos lied when he said that I did it.If the authorities also know that exactly one of the four suspects is telling the truth, who didit?1. Let p, q, r, and s be propositional variables. Which of the following expressions are NOT correct formulas of propositional logic? Select all that apply. p ¬ ∧ q ∧ r p ∧ (q, r) q ∨ s ∨ q q¬r 2. Consider the propositions p: You get an A on the Logic courseq: You pass the Algorithms course. How would you write in propositional logic the natural language statement You don't get an A on the Logic course and you pass the Algorithms course ¬p → q ¬p ∧ q p ∧ ¬q q → ¬p 3. The formula p → (p ∨ q) is _____. true for two interpretations and false for the rest valid satisfiable but not valid contradictory 4.The number of interpretations of a formula with 6 propositional variables is ___. 5.The formula (p ∨ q) ∧ r and the formula _____ are logically equivalent. (p ∧ r) ∨ (q ∧ r) (p ∨ r) ∧ (q ∧ r) p ∨ (q ∧ r) (p ∨ r) ∧ (p ∨ r) 6. Determine the equivalence law used in the expression; p and q denote…If U= {a,e,i,o,u} and A={a,c,e} and B={g,o,d}, what is A^c - B^c?
- The language we are using here is in Racket. Please try tying each of the following (car '(11 12 13 14)) (car '(a b c d)) (cdr '(11 12 13 14)) (cdr '(a b c d)) (car (11 12 13 14)) (cdr (a b c d)) What do car and cdr do? That last line of the above should cause an error. Explain why the single quote necessary and what it does.For the given statement below, “ For any two integers a and b, if both ab and a+b are even, then both a and b are even.” Do the following three things (Assume that E represents the even number set and O represents the odd number set) : Translate the following statement into a symbolic logic form Negate the logic form you came up with from step a), and simplify the negated form as much as you can. Give the contraposition of the logic form from step a), and translate the contraposition back to plain English (Hint: No need to change the quantifiers if any. Why?).1.Translate the symbols into words using the following representations:P: Dianne is a college freshman.Q: Dianne is a class officer.a.P v Q b. P ^ Q c. ~P ^ Q d. P v ~Q e. P ⇔ Q2.Translate the words into symbols using the following representationsP: Jasmine is a student.Q Jasmine is a Filipinaa.Jasmine is a student and she is a Filipina.b.Jasmine is a student and she not is a Filipina.c.Jasmine is not a student nor is she a Filipina.d.If Jasmine is a student, then she is a Filipina.e.If Jasmine is not a Filipina, then she is not a student.3.Given the Conditional Statement. “If the angles in a triangle measures 450 – 450 -900, then it is an isosceles right triangle. Construct a statement as to the following:a.Conjunction b. Disjunction c. Negation d. Biconditional