Logic basic

Question

$R(x): \quad x$ is a rabbit

$H(x): \quad x$ hops

Convert the following First Order Logic expressions into English statements.

1. $\forall x: \Bigl ( R(x) \to H(x) \Bigr )$
    
2. $\forall x: \Bigl ( R(x) \land H(x) \Bigr )$
    
3. $\exists x: \Bigl ( R(x) \to H(x) \Bigr )$
    
4. $ \exists x:  R(x) \land H(x)$

Answer 1

1. $\forall x: \Bigl ( R(x) \to H(x) \Bigr ) \; = \;$ Every rabbit hops.
    
2. $\forall x: \Bigl ( R(x) \land H(x) \Bigr ) \; = \;$ All animals are Hoppy Rabbits. ( is hoppy rabbit correct ?)
    
3. $\exists x: \Bigl ( R(x) \to H(x) \Bigr ) \; = \;$ There exists a rabbit which hops.
    
4. $ \exists x:  R(x) \land H(x) \; = \;$ There exists a animal which is like Hoopy rabbit. Or, There exists a hoppy rabbit animal.

Comments

Your (i), (ii) are correct.

(iii) Nopes. Break that implication down into simpler terms. $x \to y \equiv \overline{x} + y$. Thus, (iii) is same as saying: "Either there exists something that is a hoppy rabbit, or it could be the case that there exists something which is not a rabbit at all."

(iv) Correct. However, just keep the precedance of operators in your mind. The precedence often stings students and causes them to lose lots of marks. https://en.wikipedia.org/wiki/First-order_logic#Notational_conventions

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Fine . But i have a doubt 

 

 Here x is R(x) 
and y is H(x) 

x-->y = x' + y 
So this should be Either there exists a animal which is not rabit or There exists a animal which hops .

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Both are the same..

1 - Either there exists a non-rabbit. Or something hops. (What you said)

2 - Either there exists a non-rabbit. Or, (if everything is a rabbit, then) there is atleast 1 rabbit which hops.

3 - Either there is a hoppy rabbit. Or there exists a non-rabbit (What I said)

:)

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Oh yes i am sorry . Thank you :)