Are there differenttypes of proofs?
There are several different types of proofs
in mathematical logic. Direct proofsare
based on rules that result in one true
proposition from two propositions. They
show that a given statement is true by
simply combining existing theorems with
or without some mathematical manipula-
tions. For example, if you have two sides
of a triangle with the same length, a defi-
nition and theorem show that a line
bisecting their vertex produces two con-
gruent triangles—a direct proof that the
angles at the other two vertices have the
same size.In logic, indirect proofs are also
called “proofs by contradiction,” and are
known in Latin as reductio ad absurdum
(“reduced to an absurdity”). This type of proof initially assumes that the opposite of
what you are trying to prove is true; from this assumption, certain conclusions can be
drawn. One then searches for a conclusion that is false because it contradicts given or
known information. Sometimes, a given piece of information is contradicted, which
shows that, since the assumption leads to a false conclusion, the assumption must be
false. If the assumption is false (the opposite of the conclusion one is trying to prove),
then it is known that the goal conclusion must be true. All of this has therefore been
shown “indirectly.”Finally, a disproofis a single instance that contradicts a proposition. For example,
the disproof of “all primes are odd” is the true statement “the number 2 is a prime and
not odd.” If a disproof exists for a proposition, then the statement is false.What are deductionand inductionand how are they used in mathematics?
Deductionin logic is when conclusions are drawn from premises and syllogisms (for
more information on these terms, see above). In this instance, a deduction is a form of
inference or reasoning such that the conclusion is true if the premises are true; or,
based on general principles, particular facts and relationships are derived. Deductive
logic also means the process of proving true statements (theorems) within an
axiomatic system; if the system is valid, all of the derived theorems are considered
valid. For example, if it is known that all dogs have four legs and Spot is a dog, we log-
ically deduce that Spot has four legs; other examples of deductive reasoning include
118 Aristotle’s syllogisms.
An example of deduction can be illustrated by our
friend Spot the dog. If Spot is a dog, and we know
that all dogs have four legs, then Spot must have
four legs. The Image Bank/Getty Images.