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
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
th
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
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
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
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),
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
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
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 set
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 co
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 n