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Expert help when you need it- Q1:3. Construct a complete truth table for the following SL sentence, determine whether or not it is truth- functionally true, truth-functionally false, or truth-functionally indeterminate, and briefly state why. (/10) (X = 0) & (X = ~0) 4. Construct a complete truth table for the following SL sentence, determine whether or not it is truth- functionally true, truth-functionally false, or truth-functionally indeterminate, and briefly state why. (/10) ~D> [(DVF) >F] 5. Construct a complete truth table for the following SL sentence, determine whether or not it is truth- functionally true, truth-functionally false, or truth-functionally indeterminate, and briefly state why. (/10) ([(HI) & (I>J)] & H) & ~JSee Answer
- Q2:Class Activity 6/28/23 MTH 231, done in class Wednesday, 6/28/23 1. Let r be the proposition "It is raining," s the proposition "The sun is shining," and w the proposition "It is windy." Write the following propositions using r, s, and w and logical connectives. (a) It is windy, and either the sun is shining or it's raining. (b) If it's not raining, then the sun is shining but it's not windy. (c) If it's raining or windy, then the sun is not shining.See Answer
- Q3:2. Let r be the proposition "It is raining," s the proposition "The sun is shining," and w the proposition "It is windy." Express each of the following compound propositions as an English sentence. (a) w V (r^8) (b) r→ (-8 ^ w) (c)¬8 → (w Vr)See Answer
- Q4:+ A ALEKS-Ahkia Holloway - HW 3.1 X G Write the statement in symbols. L x s.com/alekscgi/x/Isl.exe/1o_u-IgNslkr7j8P3jH-li0BWOJBInLRZL18Trqeg6PUZ9BEvQEFqlFxsjsXHHKJvMnqEfhfiubz6nu28sVwqAJLE60... Q : ch HW 3.1-3.3 Question 10 of 25 (1 point) | Question Attempt: 3 of Unlimited Check Write the statement in symbols. Let p= "Sara is a political science major" and let q = "Jane is a quantum physics major". Jane is a quantum physics major, or Sara is not a political science major. The statement, in symbols, is written as % 9 15 5 10 f6 i 48 6 & 7 AD OVO 0-0 58 f8 8 $ 0 0 0 fg hp ( 9 ((( f10 16 ▷II Ⓒ2023 McGraw Hill LLC. All Rights Reserved. O E ✓ 18 f11 Save For Later ✓ 19 Terms of Use f12 Ahkia ins Espand 8 prt sc © E₂ Submit Assignment K Privacy Center Accessibility 91°F Windy delete backspSee Answer
- Q5:x A ALEKS - Ahkia Holloway - HW 3.1 x Ⓒ Write the statement in symbols. x + aleks.com/alekscgi/x/Isl.exe/1o_u-IgNslkr7j8P3jH-liOBWOJBInLRZL18Trqeg6PUZ9BEVQEFqlFxsjsXHHKJ ||| Question 14 of 25 (1 point) | Question Attempt 4 of Unlimited T Construct a truth table for ~ 9 →p. Use T for true and F for false. P 9 -9--P 0 O B IL @ check 9 T T 10 3 (3 (9 ( -0 0-0 0-0 CAS 40 14 16 16 CANET MEGEN HULICSee Answer
- Q6:ALEKS - Ahkia Holloway - HW 3.1 X GWrite the statement in symbols. 1 X + n/alekscgi/x/Isl.exe/1o_u-lgNslkr7j8P3jH-li0BWOJBInLRZL18Trqeg6PUZ9BEvQEFqlFxsjsXHHKJvMr ||| HW 3.1-3.3 Question 15 of 25 (1 point) | Question Attempt: 1 of Unlimited Construct a truth table for (p ✓ q) ^ (g ^ p). Use T for true and F for false. Pq (pvq) ^ (g ^p) L T IL F TI T 9 Check LL D 0 D 21 20 19 -0 0-0 0-0 9 58 O HINT hp 15 © 2023 McGraw Hit LLC. All Rights ResSee Answer
- Q7:leks.com/alekscgi/x/Isl.exe/1o_u-IgNslkr7j8P3jH-liOBWOJBInLRZL18Trqeg6PUZ9BEVQEFqlFxs search ||| HW 3.1-3.3 Question 16 of 25 (1 point) | Question Attempt: 2 of Unlimited 1 IL • P 9 7 9 → p) ~ p) ~ ~r U F 11 Construct a truth table for (9 → p) ✓ ~ 7. Use T for true and F for false. II IL F Checkk L IL IL T 11 L T UL T 9 F HW 3.1 X GWrite the statement in symbols. L X + F F Af 10 G ☐ 0 [ U 1 - 11 ✓ 12 -0 0-0 0-0 DvC DAD X ✓ 13 58 14 © 2023 McGrawSee Answer
- Q8:A ALEKS-Ahkia Holloway - HW 3.1 x Ⓒ Write the statement in symbols. L x | + om/alekscgi/x/Isl.exe/1o_u-IgNslkr7j8P3jH-li0BWOJBInLRZL18Trqeg6PUZ9BEVQEFqlFxsjsXHHKJvMnqEfhfiubz III ( Question 9 of 25 (1 point) | Question Attempt: 6 of Unlimited Check 10 ✓11 The statement, in symbols, is written as I ✓12 Write the statement in symbols. Let p= "Sara is a political science major" and let q = "Jane is a quantum physics major". Sara is a political science major, and Jane is a quantum physics major. DAD OVO C-0 -0 X D 6 9 15 0 THE BENE 7 ✓18 2023 McGraw Hu LLC. All Rights Reserved.See Answer
- Q9:Exercise 4. In the definition of the nested sequence of A's in the preceding proof, we did not write: Ait1 [A₁U {4₁} A; U{-₁} if A, U {₁} is finitely satisfiable, if A, U{-₁} is finitely satisfiable. Explain why. Hint: Exhibit a set I of wff's and a single wffy such that both IU {p} and TU{-} are satisfiable.See Answer
- Q10:Exercise 17 (Cliques in Simple Graphs, III). This continues the examination of the clique problem in Example 15 and Exercise 16. We say that a set A of propositional wff's is satisfied by G iff, for every & A, it is the case that is satisfied by G. The latter notion is defined in Example 15. Let k> 1 be fixed. Show that it is possible to define an infinite set I'(G) of propositional wff's over the set of variables X (as defined in Example 15) such that for every simple graph G, finite or infinite, it is the case that I'(G) is satisfied by G iff G does not contain a k-clique. Thus, the absence of k-cliques in a graph G, finite or infinite, can be expressed by a set I(G) of propositional wff's. Note the contrast with the conclusions in Example 15 and Exercise 16, where it is pointed out that the presence of k-cliques in an infinite graph G cannot be expressed in PL, whether by a single wff or by a set of wff's. Hint: If G does not contain a k-clique, then it does not contain a k'-clique for every k'> k. You need to show that every finite subset of I'(G) can be satisfied, and then invoke Compactness.See Answer
- Q11:Exercise 21 (Vertex Coloring in Simple Graphs). All graphs G = (V, E) in this exercise are simple, finite or infinite. Once more, we take the set V of vertices as an initial fragment of the positive integers {1,2,...}, finite or infinite, and its set of edges E CV X V. Such a graph G is said k-colorable if it is possible to assign only one of k colors to every vertex of G such that the two endpoints of every edge are assigned different colors. A famous (and very difficult to prove) result of graph theory is that every finite planar graph is 4-colorable. In this exercise you are asked to show that this result can be extended to infinite planar graphs using Compactness for PL. Let k> 1 be fixed, the number of available colors. For convenience, use two separate sets of propositional variables, Q and C, instead of P: def def {9₁.j|i,j€ {1,2,...}} and C f {ci € {1,2,...} and 1 ≤ j≤ k}. All wff's in this exercise should be in WFFPL (QUC). Use variables in Q to model a given graph G: there is an edge connecting two distinct vertices i and jiff qij is set to truth value true. Use variables in C to model G's coloring: vertex i is assigned color j = {1,...,k} iff is set to truth value true. There are three parts in this exercise, the first two are not limited to planar graphs: 1. Let G = (V, E) be a finite simple graph. Write a wff (G) which is satisfied iff G is k-colorable. Hint: Define (G) in two parts, one part specifies the structure of G (including conditions that there are no self-loops and that G is undirected), and one part specifies that G is k-colorable. You will find it useful to review Example 20. 2. Let Gf (V, E) be an infinite simple graph, with V = {1,2,...}. Show that G is k-colorable iff every finite subgraph of G is k-colorable. Hint: If an infinite G is k-colorable, then so is every finite subgraph of G. This is the easy implication. Compactness for PL should give you the converse, after writing an infinite set Þ(G) of wff's in WFFPL (QUC) expressing the k-colorability of G. 3. Let G = (V, E) be an infinite planar graph, with V = {1,2,...}. Invoke the preceding part to conclude that G is 4-colorable.See Answer
- Q12:Exercises 1.5 1. Show that a formula is valid iff T = 0, where T is an abbreviation for an instance p V -p of LEM. 2. Which of these formulas are semantically equivalent to p → (q V r)? (a) qv (-p Vr) (b) q^-r-p (c) p^rq * (d)¬q^r-p. 3. An adequate set of connectives for propositional logic is a set such that for every formula of propositional logic there is an equivalent formula with only connectives from that set. For example, the set {-, V} is adequate for propositional logic, because any occurrence of A and → can be removed by using the equivalences → ¬V and o^= -(-V¬). (a) Show that {¬, ^}, {¬,→} and {→,1} are adequate sets of connectives for propositional logic. (In the latter case, we are treating as a nullary con- nective.) (b) Show that, if C C {¬, ^, V, →, 1} is adequate for propositional logic, then ¬EC or LEC. (Hint: suppose C contains neither nor and consider the truth value of a formula o, formed by using only the connectives in C, for a valuation in which every atom is assigned T.) (c) Is {,} adequate? Prove your answer.See Answer
- Q13:Exercise 8. We can restrict the logical connectives to {, V, A}. The set of propositional vari- ables is again P = {po,P1, P2,...}, which is countably infinite. Suppose we extend this syntax with two new connectives, denoted W and M, each taking as a single argument a countably infinite set of previously defined wff's. The resulting syntax is one version of Infinitary PL. If I is the countably infinite set {91, 92, 93,...), then WI can be informally understood as: Wr = 41 V 4₂ V 43 V..., and similarly for Mr. Keep in mind that "41 V 42 V 43 V..." is not a legal formal expression; we use Was a unary connective, applied to a single argument which must be a set, and similarly for . There are three parts in this exercise: 1. Define the syntax of Infinitary PL, preferably in an extended BNF (Backus-Naur Form). Try to be as precise as you can, paying special attention to the presence of ellipses "..." in the definition - or can you think of a formulation that avoids any mention of ellipses? 2. Define the semantics of Infinitary PL, by structural induction on the syntax in Part 1, starting from an assignment of truth values to every member of P (for the base case of the induction). 3. Show that Theorem 2 does not hold, and therefore nor does Corollary 7, for Infinitary PL. Hint: Define a countably infinite set I of wff's such that every finite To CI is satisfiable, but I is not. Further Hint: Include wff = W{-Po: P₁, P2,...} in your proposed r. Remark 9. The infinitary propositional logic (Infinitary PL) extends finitary propositional logic, which is just what is commonly called propositional logic (our PL here). The distinction between infinitary and finitary extends to other formal logics and their proof systems besides PL. You can take the phrase "finitary proof system" to qualify a system that generates new finite expressions (e.g., the sequents of PL in natural-deduction style or the wff's of propositional logic in Hilbert- or Frege-style) from previously generated ones by means of finitely many rules that require each finitely many premises or antecedents - without using any notion of infinite sequence or any notion of infinite set.9See Answer
- Q14:16 Exercise 27 (Queens Problem). The n-Queens Problem is the problem of placing n queens on an nxn chessboard so that no two queens can attack each other. A solution of the problem when n-6 is shown on the left of Figure 1.2. In this exercise we specify the requirements of a solution for the n-Queens Problem as a propositional wif , with one such wff for every n 4. (There are no solutions for n-2 and 3.) For convenience, we use a set of doubly-indexed propositional variables, instead of P, where the indices range over the positive integers: Q={LJE(L2}}- The desired wif, in this exercise is in WFF(Q). We set the variable to truth value trae (resp. false) if there is (resp. there is not) a queen placed in position (i, j) of the board, where we take the first index i (resp. the second index j) to range over the vertical axis downward (resp. CHAPTER 1. PROPOSITIONAL LOGIC the horizontal axis rightward); that is, i is a row number and j is a column mmber. There are four parts in this exercise: 1. Write the wife and justify how it accomplishes its task. Hint: Write as a conjunction (a) (b) (e) (d) Aqide Age, where is satisfied if there is exactly one queen in each row, is satisfied if there is exactly one queen in each column, is satisfied if there is at most one queen in each diagonal, is satisfied if there is at most one queen in each antidiagonal. Further Hist: Given any two distinct positions (is, ji) and (ia, ja) along a diagonal, it is always the case that is −j 12-ja. And if the two positions are along an antidiagonal, then it is always the case that i, +₁=₂+/ 2. Imagine now an infinite chessboard, which occupies the entire south-east quadrant of the Cartesian plane. The coordinates along the vertical and horizontal axes are, respectively, i (increasing downward) and j (increasing rightward), both ranging over the positive integers (1,2,...). In an attempt to repeat the argument in Example 20 and Exercise 21, someone once defined the set of wis {n>4), and wrote the following (in outline here): The set I is finitely satisfiable and, therefore, satisfiable by Compactness. Hence, there exists a solution of the Infinite Queens Problem, which satisfies conditions {(a), (b), (c), (d)) for all 4. 3. Your task is to define an infinite set that: What is wrong with the preceding argument? The answer is subtle and you need to be careful. (₁|k> 1} of distinct propositional weff's such (a) For every > 1 there is n1 such that satisfaction of wff 8, implies satisfaction of wif, defined in part 1 of this exercise, ie, satisfaction of 6, defines a solution of the x-Queens Problem. (b) For all >k> 1, if satisfaction of welf's By and define solutions of the n'-Queens Problem and Queens Problem, respectively, then n²>n. (e) Every finite subset ofis satisfiable. Hint: For part (a), make 8 define a particular n-Queens Problem, ie, is satisfied by exactly one truth assigment of the variables occurring in 6. For (b) and (c), read and understand the subsection entitled "A second solution of the infinite Queens Problem" in Appendix G. 4. Let be the infinite set of weff's defined in the preceding part. Use Compactness for PL to give a rigorous argument that the Infinite Queens Problem has indeed a solution. 0See Answer
- Q15:Exercise 1. Write a natural-deduction ₁ (p^q^r) → (PV¬qV¬r) This is a more general version of de Morgan's law (4). 2. Write a natural-deduction proof of the most general de Morgan's law (4): 42 = (P₁ ^^ Pn) → (P₁ VV-pn) where n > 2. proof of the following WFF:See Answer
- Q16:Question: The last question I ask in your notes is as follows: The big question: What's the main message or idea you picked up from the story? Again, this is called the central idea that you determine. Think of the title, plot, characters, literary devices, and other ideas we discussed. I expect a well-developed paragraph that discusses what you feel is the main idea AND how the story supports your interpretation. Story: https://www.commonlit.org/en/texts/home-1 Just answer in 150 wordsSee Answer
- Q17:Problem 1 There are two parts: [Lecture Slides 10, page 18]: (a) Prove one side of the equivalence in the exercise in [Lecture Slides 13, page 32], specifically Þ→ V (not V Þ), after replacing and ; by propositional variables p and q₁, for i = 1, 2, 3. (b) Prove one side of the equivalence in the exercise in [Lecture Slides 13, page 33], specifically Þ→ V (not $), after replacing , and ; by propositional variables p; and qi, for i = 1, 2, 3, but leave as a generic (i.e., unknown) wff with two free variables. In both parts, we ask you to choose a proof-theoretic, not semantic, approach. Proof-theoretically, you can choose natural deduction or also tableaux, even though in the case of tableaux we have not yet mentioned expansion rules for quantifiers in lecture (but these are easy to formulate - left to you!). OSee Answer
- Q18:Exercise 126 (Hilbert-styles vs. Natural Deduction). Read Exercise 125, without necessarily solving it, before you attempt this one. The preceding exercise proposes a strictly proof-theoretic approach to showing that a Hilbert-style proof system and a natural-deduction proof system have equal deductive power. In this exercise we consider an alternative semantic approach, which invokes Soundness and Completeness for both proof systems. Specifically, each of the two sys- tems as here presented is sound and complete relative to the standard (classical) semantics of propositional logic based on Boolean algebras. There are three parts in this exercise, the first two of which are just plans in outline to prove the equivalence of the two systems: 1. If I, then I by Soundness of the Hilbert system. By Completeness of the natural deduction system, it follows that I END 4. 2. If I END 4, then I by Soundness of the natural deduction system. By Completeness of the Hilbert system, it follows that I ₁9. Provide the details of the two preceding parts, given only in outline here, pointing out missing prerequisites (e.g., we do not prove Soundness and Completeness for the Hilbert system in these notes) and propose ways of filling the gaps and how to prove them. (We do not add subscripts "H" and "ND" to "", because the two systems are sound and complete relative to the same semantics.) 3. Compare and discuss the pros and the cons of the proof-theoretic approach in Exercise 125 and the approach in this Exercise 126 which makes a detour through semantics. 0See Answer
- Q19:Exercise 144. Let R be a binary relation symbol and f a unary function symbol. def 1. Show that the sentence of Vr R(x, f(x)) → Vry R(x, y) is valid. Do it in two different ways: (a) proof-theoretically, p, using natural deduction, and (b) semantically, = 4. def 2. Show that the sentence Valy R(x,y) → Vr R(x, f(x)) is not valid. Note that is just the converse implication of . Hint: Try a semantic approach, i.e., show. You need to define a structure A so that the left-hand side of "" in is true in A but the right-hand side of "" is false in A. 3. Conclude that Vrly R(x, y) and Vr R(x, f(x)) are not equivalent first-order wff's. Remark: Despite the conclusion in part 3, Proposition 142 asserts Vray R(x, y) and Vr R(x, f(x)) are equisatisfiable, i.e., if there is a model for one, then there is a model for the other, and vice- versa. 42 Review the definition of o[a] in Section B.3.See Answer
- Q20:36 The next proof makes explicit reference to notions in topology. As indicated in the preceding alternative proof, a truth assignment can be denoted by a path in the full binary tree Tall, now viewed as an w-sequence in the product space (T,F). (We write for the first infinite ordinal, which is the set of natural numbers listed in their standard order.) We view {T,F) as the underlying space of a product topology ({T,F)", O), thus making every truth assignment a "point" in that topology. O is a family of open sets that define the topology, which are in this case subsets of points in {T, F)~ and satisfy the usual requirements of a topology: • The empty set and the full space {T,F) are in O. • O is closed under arbitrary unions, • O is closed under finite intersections. We can define a subset UC {T,F}" to be open iff there is a finite set of indices IC such that: U = II { Ai | i €w and A; C {T,F}} where A. = {T,F) for every i Ew-I. In words, in the infinite product U = Ao × A₁ × --- × A; x--, for all but finitely many indices i it is the case that A; = {T,F). A set U C {T,F)" is closed iff it is the complement of an open APPENDIX F. ALTERNATIVE COMPACTNESS PROOFS set. In the case of the product topology, every open subset UC (T,F) is also closed, and thus called clopen. Let A be a subset of (T,F). An open covering of A is a family of open sets {U; | i I} CO such that A CU{U; i 1}. And A is said compact if every open covering of A has a finite subcovering; i.e., there exists a finite subfamily U₁₁, UU₁ of (U; | i I} such that A (U₁, UU, U...UU.). By Tychonoff's Theorem in topology, the product topology ({T,F), O) is compact, which means that every open covering of a subset of points AC {T,F) has a finite subcovering. Alternative Proof II of Theorem 2 (Compactness for Propositional Logic). Again here, we only need to consider the non-trivial implication "+": If I is finitely satisfiable, then I is satisfiable. For every propositional wffy, let A, C {T,F}" be the collection of all points/truth assignments that satisfy p. The set A, is a closed (and open) subset of {T,F}" in the topology ({T,F)",0), which follows from the fact that only mentions finitely many propositional variables. It is easy to check that, for every finite subset A of I, if A is satisfiable, then {A | A} is not empty. Hence, the family of closed sets {Apyr} satisfies the finite intersection property. Moreover, the product topology ({T.F),O) is compact, as noted above. Hence, {A, ET) is not empty, which is the desired conclusion. 0/nExercise 147. This exercise is couched in a language which is a little more familiar to computer scientists. Given a set IWFF₁(P), the binary tree T(I) induced by I' is defined in the 35 proof above. In T(I), every finite path is a finite sequence, and every infinite path is an infinite sequence, where all the entries are in {T,F). The set of all such finite sequences is denoted {T,F)", and the set of all such infinite sequences is denoted {T,F). The root node of T(I) is represented by the empty sequence e, every leaf node is connected to the root node by a finite path, and every infinite path is one that can be traversed without ever reaching a leaf node. An arbitrary binary tree can therefore be represented by a subset of {T, F}*U{T,F}", satisfying two conditions: • U is prefix-closed, i.e., for every (possibly infinite) path ₁ € {T,F)* U{T,F) and every finite path #₂ € {T,F)", if ₁ EU and ₂ is a prefix of ₁, then #₂ EU. • For every finite path = {T,F)", it holds that T U iff F EU, i.e., every non-leaf node has two immediate successors. Given an arbitrary IWFF (P), let T(T) be the subset of infinite paths in T(I): T(T) = T(r)n(T.F)". By the preceding proof, T. (T) ‡ Ø iff the set I is satisfiable. Let To, F₁, and I2, be defined as follows: To = {-P₁ ^P₂}, T₁ = {P₁ → (P₂A---A-PE) | A>2}, del 1₂ = {~-P₁ → (P²2 A---A-¬-Pk) |>2}. There are two parts in this exercise: 1. Define the sets of infinite paths T. (To), T.(₁UF₂), and T(Tour₁ UF₂) as subsets of {T,F). In your answers, use the notational conventions of regular languages and regular w-languages that we have used earlier in the presentation of this exercise. 2. For each of the three sets defined in part 1, explain how they indicate whether the corre- sponding set of wff's is satisfiable or not. You will find it useful to consult Figure F.1. 0See Answer
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