p: "Today is cold day"
so p is the propositional constant and the rest is the statement.
There are two types of proposition:
1) Simple: Example is:
p: "Today is holiday"
q: " Dad is at home"
2)Complex/Compound:
When we are combining multiple propositional statements into a single statement using logical connectives that is known as compound proposition.
p and q: " Today is holiday and Dad is at home"
over here"and" is the logical connective.
There a multiple logical connectives such as:
1)Conjunction (and) . / ^
2)Disjunction (or) +/v
3) Negation (not) ~
4) Implication(if ....then) ->
5) Equivalence/Biconditional (if and only if)
<=> or <->
1) Conjunction
------------------------
p: "Today is holiday"
q: " Dad is at home"
p^q: "Today is holiday and Dad is at home"
Truth Table
p. q. p^q
0 0. 0
0 1. 0
1 0. 0
1 1. 1
2) Disjunction
------------------------
p: "Today is holiday"
q: " Dad is at home"
pvq: "Today is holiday or Dad is at home"
Truth Table
p. q. pvq
0 0. 0
0 1. 1
1 0. 1
1 1. 1
3) Negation
------------------------
p: "Today is holiday"
~p: "Today is not holiday"
Truth Table
p. ~ p
0. 1
1. 0
4) Implication
------------------------
p: "Today is holiday"
q: " Dad is at home"
p->q: "If today is holiday then Dad is at home"
we must know the broken algebraic expression :
p->q. = ~p v q. = p' + q
Truth Table
p. q. ~p. ~pvq ( p ->q)
0 0. 1. 1
0 1. 1. 1
1 0. 0 0
1 1. 0. 1
5) Equivalence/Biconditional
----------------------------------------------
p: "Today is holiday"
q: " Dad is at home"
p<->q: "If today is holiday only if dad is at home"
we must know the broken algebraic expression :
p<->q. = p.q+p'.q'
(same bit 0 and different bit 1)
Truth Table
p. q. p<->q
0 0. 0
0 1. 1
1 0. 1
1 1. 0
Predicate and subsequent
--------------------------------------------
p->q: "If today is holiday then Dad is at home"
here p is predicate and q is subsequent
Converse
---------------
p->q: "If today is holiday then Dad is at home"
if we interchange p and q then we converse statement
q->p : "If Dad is at home then today is holiday"
Inverse
-------------
p->q: "If today is holiday then Dad is at home"
~p-> ~q: "If today is not holiday then Dad is not at home"
Contrapositive
--------------------------
p->q: "If today is holiday then Dad is at home"
then inverse
~p-> ~q: "If today is not holiday then Dad is not at home"
then apply converse
~q->~p : "If Dad is not at home then today is not holiday"
