A ALEKS-Ahkia Holloway - HW 3.1 X G Write the statement in symbols. L x
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HW 3.1-3.3
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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
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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)
+ 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 backsp
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) & ~J
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.
3. For each of the following sets of statements, provide an interpretation that shows the statements to be consistent-i.e. can all be satisfied in the same interpretation.
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. 0
Exercises 2.4 def * 1. Consider the formula of VrVy Q(g(x, y), g(y, y), z), where Q and g have arity 3 and 2, respectively. Find two models M and M' with respective environments I and I' such that MF, but M' .
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.
Easier statement: For all y = Q there exists r = Z such that (y<r or r < y + 5) or r³ +3 ≤ y. Harder statement: For all y = Q there exists r € Z such that (y<r or r < y + 5) only if r³ +3 ≤ 7. 1. You will be given two logical statements, one easier and one harder. 2. Choose one of the two statements. 3. Write out the logical statement. 4. Write it out again using logical symbols (and 33, and arrows for the harder statement). 5. Write out the negation of the statement in words and in symbols. 6. State which of the two statements (the original one or it's negation) is true. 7. Give a well-written proof.
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.