[Discrete Maths] predicate Logic

Question

Comments

C is answer ??

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yes.How did you get this?

The negation in the front should go till the last level and it will make every case as ~Female?Please explain your approach

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https://gateoverflow.in/133130/discrete-maths-first-order-logic

Answer 1

First convert this to have better understanding.
"No female like a male."
$\forall x$ $\forall y$ ( (Female(x) ^ Male(y)) $\rightarrow$ ~like(x, y) )

"No female like a male who doesn't like all vegetarian".
$\forall x$ $\forall y$ $\forall z$ ( (Female(x) ^ Male(y) ^ Veg(z) ^ ~like(y, z) )  $\rightarrow$ ~like(x, y) ).

$\forall x$ $\forall y$ $\forall z$ ( ~(Female(x) ^ Male(y) ^ Veg(z) ^ ~like(y, z) )  $\vee$ ~like(x, y) ). 
As (P $\rightarrow$ Q) == (~P v Q)
Also ~(~P) = P. So we can put double negation, it can't affect.

$\forall x$ $\forall y$ $\forall z$ ~(~ ( ~(Female(x) ^ Male(y) ^ Veg(z) ^ ~like(y, z) )  $\vee$ ~like(x, y) ) ).
From here take one negation out of quantifiers and one negation inside statement. we can get this.
 ~ ($\exists x$ $\exists y$ $\exists z$ ( (Female(x) ^ Male(y) ^ Veg(z) ^ ~like(y, z) ) ^ like(x, y)).

So ans: C

Comments

"No female like a male who doesn't like all vegetarian". 
∀x∀x ∀y∀y ∀z∀z ( (Female(x) ^ Male(y) ^ Veg(z) ^ ~like(y, z) )  →→ ~like(x, y) ). 

Can you confirm if the following two statements means the same thing asked in questions?

1. if there exists some vegetarian that a male doesn't like then no female will like her?

2. A male who does not like all vegetarians means there is some vegetarians he don't like. It doesn't mean that he do not like any of them,what we can is that he does not like atleast one of them.Here all vegetarians do not means that any vegetarion.Like i cant say that he does not like any of them.What i can conclude is he doesn't like(Hates) at least one of them .Am i correct?

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GOOD EXPLANATION

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@Hemant

What is the difference between a and c,if i take negatation inside then all "for all" becomes "there exits" and then negation of entire predicate.Arent a and c option same?

I mean to say:- 

~ (∃x∃y∃z ( (Female(x) ^ Male(y) ^ Veg(z) ^ ~like(y, z) ) ^ like(x, y)). 

can be written as

∀x ∀y ∀z  ¬ ( (Female(x) ^ Male(y) ^ Veg(z) ^ ~like(y, z) ) ^ like(x, y)),which gives a also

Am i missing something?