First Order Logic: Gate@Zeal Test series

Question

Comments

@Aditya_ option c is meaningful as implication is there if it were (for all x) (for all y) ( S(x,j) ^ S(y,j) ^ O(x,y)) then it would have been “Every John’s sister is older than all other John’s sister”. And option a I mentioned because that statement will always be true as there will be a sister who would be older than John? Is my thinking right?

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@Sunnidhya Roy formula u mentioned means “Everyone is john’s sister and every sister is older than all other”
Option (a) is incorrect because there is no relation between John’s age and his sister’s age we are comparing among sisters.

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okay @Aditya_ Thanks 

Answer 1

None of the above sentences are expressed by the predicate logic formula given. The formula states that for all values of x and y, if x is John's sister and y is John's sister, then x is older than y. This does not relate to any of the sentences given.

Comments

@rsansiya111 Yes, Does given formula means “Every John’s sister is older than all other John’s sister” ?

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Yes, that is exactly what the given formula means. It is saying that for all values of x and y, if x is John's sister and y is John's sister, then x is older than y. This means that every John's sister is older than every other John's sister. This is because the statement is using the universal quantifier "for all" and the conditional "if...then" to make a statement about all possible values of x and y.

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Here is one possible way to simplify the given formula:

∀x,y((Sx ∧ Sy ∧ Jx) → Ox > Oy)

This formula can be simplified by removing the redundant conjunction "∧ Jx" and using a more concise notation for the universal quantifier and the conditional:

∀x,y(Sx ∧ Sy → Ox > Oy)

This simplified formula can be read as: "For all x and y, if x is John's sister and y is John's sister, then x is older than y."