Simple Proof Using Venn Diagrams
Consider the Venn diagram on the right. It appears clear that if something is in A, it must be in B, as well. We can rephrase all A is (in) B as
It is also clear that anything that is not within B can not be within A, either. This statement,
is the contrapositive. Therefore we can say that
Practically speaking, this may make life much easier when trying to prove something. For example, if we want to prove that every girl in the United States (A) is blonde (B), we can either try to directly prove by checking all girls in the United States to see if they are all blonde. Alternatively, we can try to prove by checking all non-blonde girls to see if they are all outside the US. This means that if we find at least one non-blonde girl within the US, we will have disproved, and equivalently .
To conclude, for any statement where A implies B, then not B always implies not A. Proving or disproving either one of these statements automatically proves or disproves the other. They are fully equivalent.
Read more about this topic: Contraposition
Famous quotes containing the words simple, proof and/or diagrams:
“All the phenomena which surround him are simple and grand, and there is something impressive, even majestic, in the very motion he causes, which will naturally be communicated to his own character, and he feels the slow, irresistible movement under him with pride, as if it were his own energy.”
—Henry David Thoreau (1817–1862)
“When children feel good about themselves, it’s like a snowball rolling downhill. They are continually able to recognize and integrate new proof of their value as they grow and mature.”
—Stephanie Martson (20th century)
“Professors could silence me then; they had figures, diagrams, maps, books.... I was learning that books and diagrams can be evil things if they deaden the mind of man and make him blind or cynical before subjection of any kind.”
—Agnes Smedley (1890–1950)