Write first order expression for the following

Question

1. God loves everyone.
2. Only believers respect God.
3. Frida is a non-believer.
4. To get love someone, one must respect that entity.

Answer 1

$$\begin{align}L_G(x) \quad&=\quad \text{ God loves } x\\[1em] \text{God loves everyone} \quad&=\quad \forall x\; L_G(x)\end{align}$$

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$$\begin{align} B(x) \quad&=\quad x \text{  is a believer}\\ R_G(x) \quad&=\quad x \text{ respects God}\\[1em] \text{Only believers respect God } \quad&=\quad \forall x:\;\Bigl (R_G(x) \implies B(x) \Bigr )\end{align}$$

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$$\begin{align}B(x) \quad&=\quad x \text{ is a believer}\\[1em] \text{Frida is a non-believer } \quad&=\quad \lnot B({\small\text{Frida}})\end{align}$$

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$$\begin{align} L(y,x) \quad&=\quad y \text { loves } x\\ R(x,y) \quad&=\quad x \text{ respects } y\\[1em] \substack{\text{ To get love from someone,}\\\text{one must respect that entity}} \quad&=\quad \forall y,x: \Bigl (L(y,x) \implies R(x,y) \Bigr )\end{align}$$

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This question is so full of religious propaganda XD

Comments

B(x)=x is a believer; R(x) = x respects God; 

Only believers respect God =∀x:( B(x)⟹R(x) ).

ie.For all x ,if x is a believer then x respects god.

What is wrong with this? 

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$P = \forall x: \Bigl ( B(x) \implies R(x) \Bigr )$ means "All believers respect God".

$P$ will be true when every single believer respects God. P will be false if there is atleast one believer that doesn't respect God.

$Q = \forall x: \Bigl ( R(x) \implies B(x) \Bigr )$ means that "All those who respect God are believers."

$Q$ will be true when every single person that respects God is a believer. $Q$ will be false when there is atleast one person who respects God, but isn't a believer.

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Does that make it a bit more clear as to why $Q \equiv$ "Only believers respect God"?

If not, let me know. We will have to take the really lengthy, but foolproof way of explaining it in that case :)

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as a rule of thumb you can remember it as The Case where the word Only comes. so, according to it the word Only is associated with $q$ part of $\left( p \implies q \right)$. Here, it is being a believer.

To seek robustness of this rule we need to compare, what you suggest and what the actual answer is; in little more detail.

consider this $B \implies R$ (writing it casually) meaning $\neg B \lor R$ which says that either a person is not a believer OR the person respects god; Then the entire statement will be true.
so, this statement will be true if a person respects god and is not a believer. This contradicts the given statement : Only Believers Respect God.

Apply this to the correct quantified proposition, given above and you'll see what is required to understand.

Answer 2

God loves everyone- L(ge)

Only believers respect God- ]x( B(x)^ R(xg)) ( where ] is there exist, V if for all )

Frida is a non-believer-        -B(f)  (where - is not)

To get love someone, one must respect that entity=   Vx  ]y  (Lyx  ->Rxy)

Comments

whether God Loves everyone is $\forall x L(god, x)$

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in my opinion forall x L(god, x) means god loves a particular domain x. That means god loves for all people who in the x option only.