What's the Difference Between Implication and Conjunction in Logic?

[ych22]

"Every student in this class has studied calculus".

Q(x): x is in this class.
P(x): x has studied calculus.

How come we have Q(x)->P(x) but not Q(x) ^ P(x)? What really is the difference?

 

[SW VandeCarr]

"Every student in this class has studied calculus".

Q(x): x is in this class.
P(x): x has studied calculus.

How come we have Q(x)->P(x) but not Q(x) ^ P(x)? What really is the difference?

The difference is that one is an implication and the other is a conjunction. The conjunction simply says P and Q which are one of your premises and your conclusion. The implication says Q implies P. If the implication is true, as it is given the premises, does P therefore imply Q?