17. Prove or disprove that (p →q) V (~p→q) is logically equivalent to q.18. Determine whether the given propositional forms are logically equivalent.a. p (qr) and (p q) ^ (pr)b. (pq) →r and p→ (q→r)c. ~(pq) and pqd. ~ (p V q Vr) and~p^~q^~ re. (pvq-r) V (p V~q^r) and p ^ (qr)(~P→Q) ^ (~SVR)b. P→~(QV~R)S200STIP→ (~Q^~~R)dnsmart12*-(049)19. For each of the following arguments, state the Rule of Inference by which its conclusion followsfrom its premises:2V0-a. (~P→Q) ^ (RV~S)(149) 0:11VO8(9-)/(VO(12) VO(1^2)+9

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Answered: 17. Prove or · disprove that (p → q) ✓…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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E. Establish the validity of the arguments by completing the formal proofs by supplying explanations for each step. [(p → r) a (-p → q) ^ (q → s)] → (~r →s) Steps Reasons 1. p→r 2. -r → -p 3. ~р — q 4. -r →9 5. q→s 6. :.~r →s
5) Let the domain of discourse be the set of real numbers. Indicate which propositions are true and which are false. Negate each of the following propositions. a)Vx3y(3x+ y = 4) b)ayax(x- y = 2, andx • y = 36) c)Vx(x = 1) → (x' = 1))
36. Consider the argument: Premise 1: p→q Premise 2: ~r Premise 3: qvr Conclusion: r Write the above argument in its symbolic statement form. A. {[(p→q^~r ^(qvr )]}→r B. {p→ [[q ^ ~r) ^lgvr )]→r} C. {(p→q)^[l~rn q)vr ]}→r D. {[(p→q)^~r] ^(qvr )}→r
Essentials of DISCRETE MATHEMATICS
6. Match each logical statement in the left hand column with an equivalent logical statement in the right hand column. Logical Statement (i) ~ WA~y Logical Statement (v) w v ~y (ii) wA~y (vi) ~ (~w→ y) (iii) w v y (vii) ~ (~w v y) (viii) ~ w → y (iv) ~ (~WAy)
2. Either show that (P → Q) → R and ~R → (P ^ ~Q) are logically equivalent or demonstrate that they are not.
1. Given that p and q are True and r and s are False, find the truth value of the statement. (~r) = (pA(r → (~qV~s))) 2. (а) Rewrite the statements in the form of " i. x, No irrational numbers are integers. The number -4 is not equal to the square of any real number. ii. (b) Rewrite the statements in the form of "3 i. ii. x, Some exercises have answers. Some real numbers are rational. (c) Determine the truth value of the statements where D = E = {-9, –3,0,3,9}. Hence, write the negation of the statements. i. i. x in D, 3y in E such that x +y = 0. 3x in D such that Vy in E, x + y = y. 3. Find the union, intersection, and difference (A - B) of the following pairs of sets. (a) A = The set of positive odd integers less than 15. B = The set of positive even integers less than 16. (b) A = (f,l,o,w,e,r} B = {c,l, o,n, e}
D. Determine the truth value of each proposition, given that p and q are true propositions and r and s are false propositions. Show your solution. (4 points) 1. [[(p → r)^(q → r)] → [(sVq)/^~r]] → s 2. -(~qAp) → [(p → ~s) → ~r] 3. [(pVp) → ~r]^~[(s → ~q)^~p] 4. (p + ~q) –→r + [[p → ~s) +→ ~r]
7
Match each of the following compound propositions with the logically equivalent one. (pV¬q) P→→q (p → a) (p^(-(p^ q))) 1. ¬p/q 2. p V q 3. pVq 4. p -q

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17. Prove or · disprove that (p → q) ✓ (~p → q) is logically equivalent to q.