Proof of Validity of Argument Example 5
Example 5
Everyone has either a brother or a sister.
Everyone either does not have a sister or they are not married.
Anyone who has a child is married.
Someone does not have a brother.
) There is somebody who does n
Methods of Proof
A theorem is a statement s that can be shown to be true. A proof is a valid
argument that
establishes the truth of a theorem. The statement s usually has the form p ! q,
where p is
the hypothesis (it can consist of several premises, p p1^
Images of Set Unions and Intersections
Theorem: Let f : A B be a function. If S A and T A, then
1. f (S T ) = f (S) f (T ), and
2. f (S T ) f (S) f (T ).
Proof:
1. We prove equality by showing f (S T ) and f (S) f (T ) are subsets of each other.
Suppose y
Horse of a Different Colour
Horse of a different colour is an idiomatic expression
that means an unrelated or irrelevant matter; something
that has no bearing on the discussion or question.
Horse of a different colour from Wizard of Oz
Hungarian mathemati
Mathematical Induction
Let P(n) be a propositional function of a positive integer n. The following
arguments are all valid (by using Modus Ponens and conjunction repeatedly).
P(1)
) P(1)
P(1)
P(1) ! P(2)
) P(1) ^ P(2)
P(1)
P(1) ! P(2)
P(2) ! P(3)
) P(1) ^
One-to-One and Onto
Definition: Let f : A B be a function. Then f is said to be
1. one-to-one or injective if
x1 A x2 A (f(x1) = f(x2) (x1 = x2),
or equivalently,
x1 A x2 A (x1 6= x2) (f(x1) 6= f(x2),
2. onto or surjective if
y B x A (f(x) = y),
or equiva
Permutations and Combinations
Permutations
Given a set of n distinct objects, a permutation is an ordered arrangement of these n
objects.
An r-permutation (where 0 r n) is an ordered arrangement of r elements of the set.
The number of such r-permutations
Prove that all odd positive integers greater than one are prime.
Mathematician
3 is prime
5 is prime
7 is prime
9 is not prime counter-example
the proposition is false
Physicist
3 is prime
5 is prime
7 is prime
9 is an experimental error
11 is prime
13 i
Set Operations
Denition Let A be a set contained in some universal set U . Then the complement of A (relative
to U ), denoted by A (or A or Ac ), is dened by
A = cfw_x|x A.
Denition Let A and B be sets contained in some universal set U . Then their inters
Pigeonhole Principle
Pigeonhole Principle: Let k Z+. If k + 1 or more objects are placed into k
boxes,
then at least one box must contain two or more objects.
Proof (by contraposition): Suppose the conclusion were false. Then each of
the k
boxes has at mo
Product and Sum Rules of Counting
Product Rule: Suppose a procedure can be broken down into a sequence of two tasks.
If there are n1 ways of doing the first task and for each of these ways there are n2 ways
of
doing the second task, then there are n1 n2 w