X [(∀z z|x ⇒ ((z=x) ∨ (z=1))) ⇒ ∃w (w > x) ∧ (∀z z|w ⇒((w=z) V (z=1)))]LET, X ≡ (∀z z|x ⇒ ((z =x) ∨ (z=1)))Y ≡ ∃w (w > x)Z ≡ (∀z z|w ⇒ ((w = z) ∨ (z=1))∀x [X ⇒ Y ⇒ Z]X is true: If x is prime numberY is true: If there exist a w which is greater than xZ is true. If w is primeS1 = {1,2,3, ... , 100}Let x = 97z = 97∴ z|x ⇒ (z = 1) ≡ T ⇒ T ≡ Tprime number greater than 97 is 101But S1 is not included in set. HENCE there doesn’t exist w which is greater than xT ⇒ F.Z ≡ T ⇒ F ≡ FTherefore, S1 does not satisfy φS2. Set of all positive integers satisfy φS3. Set of all integersIf x is negative, X ≡ TF ⇒ Y.Z ≡ TAlso, set of all positive integers satisfy φTherefore, OPTION 3 is correct.

"> X [(∀z z|x ⇒ ((z=x) ∨ (z=1))) ⇒ ∃w (w > x) ∧ (∀z z|w ⇒((w=z) V (z=1)))]LET, X ≡ (∀z z|x ⇒ ((z =x) ∨ (z=1)))Y ≡ ∃w (w > x)Z ≡ (∀z z|w ⇒ ((w = z) ∨ (z=1))∀x [X ⇒ Y ⇒ Z]X is true: If x is prime numberY is true: If there exist a w which is greater than xZ is true. If w is primeS1 = {1,2,3, ... , 100}Let x = 97z = 97∴ z|x ⇒ (z = 1) ≡ T ⇒ T ≡ Tprime number greater than 97 is 101But S1 is not included in set. HENCE there doesn’t exist w which is greater than xT ⇒ F.Z ≡ T ⇒ F ≡ FTherefore, S1 does not satisfy φS2. Set of all positive integers satisfy φS3. Set of all integersIf x is negative, X ≡ TF ⇒ Y.Z ≡ TAlso, set of all positive integers satisfy φTherefore, OPTION 3 is correct.

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Consider the first order predicate formula ϕ:∀x [(∀z z|x ⇒ ((z = x) ∨ (z = 1))) ⇒ ∃w (w > x) ∧ (∀z z|w ⇒ ((w = z) ∨ (z = 1)))]Here ‘a|b’ denotes that ‘a divides b’, where a and b are integers. Consider the followingsets:S1. {1,2,3, ... , 100}S2. Set of all positive integersS3. Set of all integersWhich of the above sets satisfy φ?

Logical and Verbal Reasoning Logic in Logical and Verbal Reasoning . 6 months ago

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∀X [(∀z z|x ⇒ ((z=x) ∨ (z=1))) ⇒ ∃w (w > x) ∧ (∀z z|w ⇒((w=z) V (z=1)))]LET, X ≡ (∀z z|x ⇒ ((z =x) ∨ (z=1)))Y ≡ ∃w (w > x)Z ≡ (∀z z|w ⇒ ((w = z) ∨ (z=1))∀x [X ⇒ Y ⇒ Z]X is true: If x is prime numberY is true: If there exist a w which is greater than xZ is true. If w is primeS1 = {1,2,3, ... , 100}Let x = 97z = 97∴ z|x ⇒ (z = 1) ≡ T ⇒ T ≡ Tprime number greater than 97 is 101But S1 is not included in set. HENCE there doesn’t exist w which is greater than xT ⇒ F.Z ≡ T ⇒ F ≡ FTherefore, S1 does not satisfy φS2. Set of all positive integers satisfy φS3. Set of all integersIf x is negative, X ≡ TF ⇒ Y.Z ≡ TAlso, set of all positive integers satisfy φTherefore, OPTION 3 is correct.

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Posted on 25 Oct 2024, this text provides information on Logical and Verbal Reasoning related to Logic in Logical and Verbal Reasoning. Please note that while accuracy is prioritized, the data presented might not be entirely correct or up-to-date. This information is offered for general knowledge and informational purposes only, and should not be considered as a substitute for professional advice.

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