Qualifiers in a Formal Mathematical Proof

When writing a mathematical proof, we often want to make a statement that applies to a collection of numbers or objects. To do this, we often need to add qualifiers to our statements which describes precisely which objects we are referring to in our statement and how many of them this applies to.

For example, suppose we are considering the proposition

For all $n \in \N$ the square of $n$ is greater than or equal to $n$

In making this statement, we are making a broad claim about all values of $n \in \N$ , and it would be untrue if we could find a single value of $n$ for which this statement did not hold.

Another proposition we could consider is:

A value of $x \in (0, \pi)$ exists such that $x = 2\sin{x}$

For this statement to be true, we only need it to be true for a single value of $x \in (0, \pi)$ . If there were multiple values, the statement would still be true, the only way this statement could be false is if there are no values of $x$ which satisfy this property.

In the above two propositions, the use of a qualifier specifies how many values the proposition is applicable to:

The 'for all' qualifier (written as $\forall$ ) is used to specify that the proposition applies to every single value in the context of the proposition.
The 'there exists' qualifier (written as $\exists$ ) is used to specify that the proposition applies to at least one value in the context of the proposition, without specifying which particular value.

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