Three important kinds of statements in mathematics:
- A universal statement is true for all elements in a set, e.g. “all positive numbers are greater than zero”
- A conditional statement is true when something else is also true, e.g. “if 200 is greater than 100, then 200 is greater than 99”
- An existential statement is true for at least one thing where a property could be true or false, e.g. “there is a prime number that is even”
Statements can be combined:
- Universal conditional statements - for every fruit a, if a is a blueberry, then a is a berry.
- Universal existential statements - for every a there exists b, e.g. for every positive integer there exists an additive inverse
- Existential universal statements - there exists an a which satisfies a universal statement, e.g. there is a positive integer a that is less than or equal to every positive integer