Transcript File

two
Biconditionals
A biconditional is a combination of two statements;
a conditional (Should be True)
its converse (Should be True)
How to combine them?
If p, then q AND if q, then p
p if and only if q.
or
q if and only if p.
Has to be true
Write the conditional and its converse as a biconditional
Conditional:
If a figure has eight sides, then it is an octagon. True
Converse:
If a figure is an octagon, then it has eight sides. True
 Biconditional:
A figure is an octagon, if and only if it has eight sides.
Write the conditional and its converse as a biconditional
Conditional:
If three points are collinear, then they lie on the
same line. True
Converse:
If three points lie on the same line, then they are
collinear. True
 Biconditional:
Three points are collinear, if and only if they lie on the
same line.
Separate the biconditional into its parts
A number is divisible by 3 iff the sum of its digits is
divisible by 3
If a number is divisible by 3, then the sum of its digits
is divisible by 3
If the sum of a number’s digits is divisible by 3, then
the number is divisible by 3
Separate the biconditional into its parts
A number is prime iff it has only two distinct factors 1,
and itself
If a number is prime, then it has only two distinct
factors 1, and itself.
If a number has only two distinct factors 1, and itself,
then it is a prime number.
Definitions
* A good definition is precise  AVOID words
such as large, sort of, almost, …
* A good definition uses clearly understood
terms. The terms should be commonly
understood or already defined.
* A good definition is reversible  can be
written as a true biconditional.
Good or bad Definition ?
* A triangle has sharp corners
BAD:
- “sharp” is imprecise
-
It is not reversible.


Conditional: If a shape is a triangle, then it has
sharp corners TRUE
Converse: If a shape has sharp corners, then it is a
triangle. FALSE (counterexample: star)
Good or bad Definition ?!
* Perpendicular lines are two lines that intersect
to form right angles Good Definition
Test if reversible:


YES reversible
Conditional: If two lines are perpendicular, then
they intersect to form right angles. True
Converse: If two lines intersect to form right
angles, then they are perpendicular True