Special Binary
Relations
Special relations
Definition. A binary relation on a set is reflexive if for
all , .
Example. Let . Consider the following binary relations
on :
1.
2.
3.
Among the three given
At the end of the period, you should be
able to:
Show logical equivalences of conditional
or biconditionals statements. (CO1,
CO2)
Note: If p and q are logically equivalent,
the symbol p q (p is equ
LOGO
Introduction to Proofs
LOGICAL PROOFS
A mathematical system consists of
Undefined terms
Definitions
Axioms
Theorems etc.
LOGICAL PROOFS
Undefined terms are the basic building blocks
of a math
LOGO
Predicates and Quantifiers
Predicates
The sentence x + 2 = 2x is not a proposition,
but if we assign a value for x then it becomes a
proposition.
The phrase x + 2 = 2x can be treated as a
functi
LOGO
Logical Equivalences and
the Rules of Replacement
RULES OF REPLACEMENT
1. Idempotent Laws
pp p
pp p
2. Transposition
(pq) (q p)
3. Double Negation Law
(p) p
4. De Morgans Laws
(pq) (p q)
(p
Mathematical Induction
A standard procedure for establishing
the validity of mathematical statements
which we shall call as propositions
involving a series of positive integers.
Mathematical Induction
FUNCTION
As a Rule
As a Set of Ordered Pairs
Concept of function
We may imagine a function to be some sort of a box or
machine. When something is fed into the machine, there
will be an output. The out
LOGIC
1
Truth table
A proposition may consist of other propositions. The truth
value of this proposition may change depending on the truth
values of the other propositions it is made of. The truth
val
Equivalence
Relation
Reflexive, Symmetric, and Transitive
Definition. A binary relation on a set is called an
equivalence relation if it is reflexive, symmetric, and
transitive.
Example. Let and
The t
LOGIC
Predicates and Quantifiers
Let be a non-negative integer. An -place predicate is a
statement containing variables. By fixing the values of the
variables in a predicate, it becomes a proposition.
BINARY RELATION
Definition. A binary relation on a set is any subset of .
Example. Let . Then
, (2,1), (2,2), (2,3), (3,1), (3,2), (3,3)
By definition, a binary relation on is any subset of .
Therefor
Principle of Inclusion
and Exclusion
Principle of Inclusion and Exclusion
Let , , , be arbitrary sets. Then
Examples
Exercise. Write the formula for using the principle of
inclusion and exclusion.
Ill
ORDERING
RELATIONS
PARTIAL ORDERING
LINEAR ORDERING
Let us first discuss a new and more convenient notation for
membership in a binary relation. If is a binary relation on a
set , then the statement w
At the end of the period, you should be able to:
Identify propositions from non-propositions.
(CO1, CO2)
Perform operations on simple or compound
propositions using logical operators.
(CO1, CO2)
Fi