Translating a sentence into Symbolic Notation Let's say for example,
Every son of Trump has a mother but not every son of Trump has the same mother where Pr: x is a person; Fx: x is male; Pxy: x parents y; Tx: x is current Pres of the USA. 
How would I go about this translation? I have not yet studied symbolic logic, but am interested and planning on it since I have a background in computer science
 A: Let's make a predicate Mxy for "y is x's mother". Well, a mother isn't male, and is a parent. So $Mx \equiv Pxy \land \neg Fy$.
How about Sx for "x is a son of Trump"? Well that means x is male, there exists a person who is his parent, and that parent is Trump. So that's $Sx \equiv Fx \land \exists y(Pxy\land Ty)$.
So let's look at "Every son of Trump has a mother". This is saying that, for any person, if that person is a son of Trump, then there is a person who is that person's mother. Symbolically, this is $\forall x[Sx\rightarrow\exists y (Mxy)]$. Expanding out the predicates gives $\forall x([Fx \land \exists z(Pxz\land Tz)]\rightarrow\exists y [Pxy \land \neg Fy])$.
Now let's look at "every son of Trump has the same mother". That means there is a person y such that for every person x, if person x is a son of Trump, then person y is person x's mother. Symbolically, this is $\exists y[\forall x(Sx\rightarrow Mxy)]$, and expanding out the predicates again gives $\exists y[\forall x([Fx \land \exists z(Pxz\land Tz)]\rightarrow [Pxy \land \neg Fy])]$.
Lastly, the whole statement "Every son of Trump has a mother but not every son of Trump has the same mother" is just saying "[Every son of Trump has a mother] $\land\neg$[every son of Trump has the same mother]", so the full logical statement is
$$
\forall x([Fx \land \exists z(Pxz\land Tz)]\rightarrow\exists y [Pxy \land \neg Fy])\land \neg(\exists y[\forall x([Fx \land \exists z(Pxz\land Tz)]\rightarrow [Pxy \land \neg Fy])])
$$
If you want to practice manipulations on symbolic first-order logic, try simplifying this.
