Universal Statement is a #202204281244 which states that certain properties are true for all elements in a set. It usually uses universal quantifier \(\forall\) which express for all, for every, for arbitrary, for any, for each and given any.
For example: All dogs are mammal can be denoted in mathematics as \(\forall x \in H, x \text{ is mammal}\) where \(H\) represents all dogs, which read For all \(x\) in the set of all dogs, \(x\) is mammal.
To disprove this, one could use counterexample, to show that at least one \(x\) in the set of \(H\) is false. To prove that the statement is true, one could not avoid proving that all of them are true, which is called the method of exhaustion. You can see the difference between it and 202204281254.
The negation of a Universal Statement is logically equivalent# to an 202204281254. Formally, this means that \(\sim (\forall x \in D, Q(x)) \equiv \exists x \in D \text{ such that } \sim Q(x)\). For instance, the negation of the statement “All books have a cover” is “There is at least one book which doesn’t have a cover”. It is tempting to say that the negation should be “No books have a cover”, but it is enough to disprove the above statement by just one counterexample.
Just like 202205062055, Universal Statement could be said is vacuously true or true by default if \(Q(x)\) is false for every element in the set. In other word, if the negation of the Universal Statement is false, then the Universal Statement is true, even if there is no element in the set. Consider the statement “All books on the shelf has a cover”. If none of them are on the shelf, then the statement is true, as the negation of it “There is at least one book on the shelf which doesn’t have a cover” is false, as there is no books on the shelf at all.