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Quantifiers
These are words or symbols which specify the quantity or existence of something in a mathematical statement.
e.g.
- All square numbers are positive.
- For all people in Australia, the equator is to the North.
- There exists an even prime numbers
- There exists: they can be infinitely many, or just one. But there has to be at least one case.
- For all real number $x > 0$, there exists a real number $z$ where $0
Symbols:
- $\forall$ mean 'for all'
- $\exists$ means 'there exists'
- Note: in assessments, don't use these symbols!
Proving/Disproving there exists a square prime number statements with quantifiers
- There exists a prime here exists a square prime number.number which is not odd.
- Proof: 2 is even, 2 is prime.
- TLDR: By citing one example is adequate to prove "There exists" statements are true
- However, you need a more rigorous proof to disprove something.
- All right triangles are isoceles.
- To prove this is true, you need a rigorous proof. Yet to prove this is false, you need just one example.
Summary
- When dealing with a "There exists" statement,
- To prove the statement is true, only one example must be provided.
- To prove the statement is false, a rigorous proof must be provided.
- When dealing with a "All/For all" statement,
- To prove the statement is true, a rigorous proof must be provided.
- To prove the statement is false, only one example must be provided.
