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

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The discussion clarifies the distinction between implication and conjunction in logic, using the example of students studying calculus. It explains that Q(x) → P(x) indicates that being in the class implies having studied calculus, while Q(x) ∧ P(x) would require both conditions to be true simultaneously. The implication suggests a one-way relationship, whereas the conjunction asserts both premises must hold. The conversation raises the question of whether P implies Q, but does not reach a definitive conclusion. Understanding these differences is crucial for logical reasoning.
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"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?
 
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ych22 said:
"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?
 
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