English to Predicate Logic - And vs Imply “Every student in this class has taken a course in Java.”
But if U is all people, also define a propositional
function S(x) denoting “x is a student in this class” and
translate as ∀x S(x)→ J(x).
But why is ∀xS(x)∧ J(x) incorrect?
S(x)→ J(x) is True when:


*

*X is a student in the class and has taken a course in Java.

*X is not a student in the class and has taken a course in Java.

*X is not a student in the class and has not taken a course in Java.


S(x) ∧ J(x) is True when:


*

*X is a student and has taken a course in Java.

 A: $∀x S(x)→ J(x)$ means when some $x$ in $U$ in this class, then he must taken Java.
$∀x S(x)∧ J(x)$ means any $x$ in $U$ must be both in this class and also taken Java.
The first statements says nothing about those $x$ in $U$ but not in that class, the second statement claimed that any $x$ are in that class, basicly thats the difference of them.
From the truth table, this would be even more clear:
$$\begin{array}{ccc}S(x)&J(x)&S(x)\to J(x)\\F&F&\boxed{T}\\F&T&\boxed{T}\\T&F&F\\T&T&T\end{array}$$
$$\begin{array}{ccc}S(x)&J(x)&S(x)\land J(x)\\F&F&\boxed{F}\\F&T&\boxed{F}\\T&F&F\\T&T&T\end{array}$$
Second one will not hold if that $x$ is not in class,
however in this case the first statement is vacuous true.
A: If the domain of discourse, or $U$, is the set of all people, then the statement $\forall x [S(x) \wedge J(x)]$ may be translated "For every person $x$, $x$ is a student in this class and $x$ has taken a course in Java," or in other words, "Every person is a student in this class and has taken a course in Java."
This is very different from the meaning of $\forall x [S(x) \rightarrow J(x)]$, which translates as "For every person $x$, if $x$ is a student in this class, then $x$ has taken a course in Java," or in other words, "Every person who is a student in this class has taken a course in Java."
Again, the first statement says all people are students of the class and have taken Java; the second statement says only people who are students of the class have taken Java. If all people in the domain of discourse are actually students of the class, then it is true that both statements yield the same truth value. However, if there is at least one person in the domain of discourse that is not a student of the class, then the first statement is false while the second statement remains true. Hopefully this sheds some light on why the two statements are not logically equivalent.
