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First of all, I have never studied Logic seriously before. I am reading this article on Wikipedia. The definition is the following:

Logical conjunction is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both of its operands are true.

I would like to know the motivation for this definition. For example, I do not understand why if $A$ is false and $B$ is true, then $ A \wedge B$ is false.

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You should think of conjunction as "and". So $A \wedge B$ is true precisely when $A$ and $B$ are true. Similarly, disjunction is an "or" operator. So $A \vee B$ is true precisely when $A$ or $B$ is true.

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The result of $A$ and $B$ is true only when (if and only if) both $A$ and $B$ are true. If (at least) one of them is false, the result will be false too.

(Logical conjunction is just 'and' and the notation for it is the one you gave: $A\wedge B$.)

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Logical conjunction is more commonly (in some circles) referred to as "and." That is, $A\wedge B$ is true iff $A$ and $B$ are true. "And" implies both (not only one) are true.

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