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logicdiscrete-mathematicsquantifiers

Translate English sentence into predicated wff

I have a school assignment that needs to translate the English sentence into predicate logic.

The question is "Every fruit that is an apple is not a mango."

The domain is the whole world, and M(x) means x is a mango, A(x) means x is an apple, F(x) means x is a fruit.

The answer I came up with is "(∀x)[F(x)∧(A(x)⟶[M(x)]′)] " But there is someone who answers it like below.

"(∀x)[F(x)⟶(A(x)⟶[M(x)]′)]"

Will the second answer be a valid translation for the sentence "Every fruit that is an apple is not a mango."?

Solution

  • Both are correct. You're essentially asking if:

    A -> (B -> C)

    is equivalent to:

    (A /\ B) -> C

    And if you tabulate the truth tables, you'll see that these two propositions are always equivalent.