Quantifier

Question

Express the statement “If a person is female and is a parent, then this person is someone’s
mother” as a logical expression involving predicates, quantifiers with a domain consisting of all
people, and logical connectives.

F(x) to represent “x is female,” P(x) to represent “x is a parent,” and
M(x, y) to represent “x is the mother of y.

how these two statements are equivalent ?pls explain

∀x((F (x) ∧ P(x)) → ∃yM(x, y)).

∀x∃y((F (x) ∧ P(x)) → M(x, y)).

Comments

@Bikram sir

Answer 1

Statement 1: for all x, if x is female and parent, then there exists some y, whose mother x is.

Statement 2: For all x, if x is parent and female the it's mother of some y.

Yes both are same statements! y is independent of x, it scope is applied on M(x,y) only, so you can bring it inside from outside.