In Exercises 17-30, either prove that the wff is a valid argument or give an interpretation...

 

 

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In Exercises 17-30, either prove that the wff is a valid argument or give an interpretation in which it is false. 17. (3x)[A(x) ^ B(x)] → (3x)4(x) ^ (3x)B(x) 18. (3x)[R(x) V S(x)] → (3x)R(x) V (3x)S(x) 19. (3x)P(x) (3x)(y)Q(x, y) → (3x)(y) [P(x) ^ Q(x, y)] 20. (Vx) [P(x)→→ Q(x)] → [(Vx)P(x) → (Vx)Q(x)] 21. (Vx)(P(x))' → (Vx)(P(x) → Q(x)) 22. [(Vx)P(x) → (V\x)Q(x)] → (Vx)[P(x) → Q(x)] 23. (3x)(Vy)Q(x, y) → (Vy)(3x)Q(x, y) 24. (Vx)P(x) V (3x)Q(x) → (Vx)[P(x) VQ(x)] 25. (Vx)[A(x)→ B(x)] → [(3x)A(x) → (3x)B(x)] 26. (Vy)[Q(x, y) → P(x)] → [(y)Q(x,y) → P(x)] 27. [P(x) → (y)Q(x, y)] → (=y)[P(x) → Q(x, y)] 28. (Vx)(P(x) VQ(x))^(3x)Q(x) → (3x)P(x) 29. (3x) [P(x) ^ Q(x)]^(Vy)[Q(y) →→R(y)] → (3x)[P(x) ^ R(x)] 30. (Vx)(Vy)[(P(x) ^ S(x, y)) → Q(y)] ^ (3x)B(x) ^ (Vx)(B(x) → P(x)) ^ (Vx)(y)S(x, y) → (3x)Q(x) In Exercises 17-30, either prove that the wff is a valid argument or give an interpretation in which it is false. 17. (3x)[A(x) ^ B(x)] → (3x)4(x) ^ (3x)B(x) 18. (3x)[R(x) V S(x)] → (3x)R(x) V (3x)S(x) 19. (3x)P(x) (3x)(y)Q(x, y) → (3x)(y) [P(x) ^ Q(x, y)] 20. (Vx) [P(x)→→ Q(x)] → [(Vx)P(x) → (Vx)Q(x)] 21. (Vx)(P(x))' → (Vx)(P(x) → Q(x)) 22. [(Vx)P(x) → (V\x)Q(x)] → (Vx)[P(x) → Q(x)] 23. (3x)(Vy)Q(x, y) → (Vy)(3x)Q(x, y) 24. (Vx)P(x) V (3x)Q(x) → (Vx)[P(x) VQ(x)] 25. (Vx)[A(x)→ B(x)] → [(3x)A(x) → (3x)B(x)] 26. (Vy)[Q(x, y) → P(x)] → [(y)Q(x,y) → P(x)] 27. [P(x) → (y)Q(x, y)] → (=y)[P(x) → Q(x, y)] 28. (Vx)(P(x) VQ(x))^(3x)Q(x) → (3x)P(x) 29. (3x) [P(x) ^ Q(x)]^(Vy)[Q(y) →→R(y)] → (3x)[P(x) ^ R(x)] 30. (Vx)(Vy)[(P(x) ^ S(x, y)) → Q(y)] ^ (3x)B(x) ^ (Vx)(B(x) → P(x)) ^ (Vx)(y)S(x, y) → (3x)Q(x)

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