example of quantifier
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Example 1: let be the property where x is a real number. is true and is false.
Example 2: a property can have two or more variables. Let be . in this case is true but is false because is not equal to .
Definition: let be a property on the set , i.e. is a property and varies in the set . a) The symbol means for every in the set the proposition is true. b) The symbol means there is some in the set for which the proposition is true. If , i.e. if the set is empty, is defined to be true and is defined to be false.
Example 1: is a true proposition.
Example 2: is false, because no real number satisfies .
Example 3: is a property. varies in . As a result is a proposition, i.e. it is a true or a false sentence. In fact is false but is true; here is the interval containing real numbers greater than .
for proofs of the following theorems see the address above
Theorem 1: if and then .
Theorem 22: if then .
here is a property on .