Double implication or biconditional
Hello friends, Welcome to my blog mathstips4u.
In my last post we have seen
implication and its truth table and some of its examples. Some of important
examples I intentionally left as exercise. The solution of these examples is
provided at the end of this post.
We shall start with
Bi-conditional or
double implication (↔):
Let p and q be two simple
statements. Then the compound statement ‘p if and only
if q’ is called the bi-conditional or double implication, denoted by p ↔q or p = ＞ q. It is read as p implies and implied by q.
p ↔q is defined to
have the truth value ‘true’ if p and q both have the same truth values.
Otherwise
p ↔q is defined to
have the truth value ‘false’.
Truth table of bi-conditional p ↔q
p
|
q
|
p ↔q
|
T
|
T
|
T
|
T
|
F
|
F
|
F
|
T
|
F
|
F
|
f
|
T
|
Note: 1. p ↔q, q
↔p both are same.
2. p ↔q is the conjunction of
a conditional and its converse i.e. p → q and q → p.
i.e. p ↔q ≡
(p → q) Ʌ (q → p)
I will prove it in my next video
on Logical
Equivalence.
In this way we have seen
double implication and its truth table.
We shall see exercise from my last post on
conditional.
Ex. Express following in
symbolic form.
2. I shall come provided I finish
my work.
3. A family becomes literate if
the women in it are literate.
4. Rights follow from performing
the duties sincerely.
5. x = 1 only if
6. The sufficient condition for being
rich is to be rational.
7. Getting bonus is necessary
condition for me to purchase a car.
|
First
we rewriting each statement using if …then
2.If I finish my work, then I shall come.
p: I finish my work. q: I shall come.
|
3.
If the women in a family are literate, then a family becomes literate.
p: The women in a
family are literate, q: A family becomes
literate.
|
4.If the duties are performed sincerely
then the rights, follow.
p: The duties are performed sincerely., q: Right follow
5. If x^2
p: x^2 = x,
q: x = 1.
6. If one is rich, then he is rational.
p: One is rich, q: he is rational.
7. If I get bonus, then I can purchase a car.
P: I get bonus, q: I can purchase a car.
The symbolic form for all the examples
from (2) to (7) is p →
q.
Observe carefully how the last two statements are
written in by using if…then.
In my next post we are going to learn Statement pattern and
Logical Equivalence and some of their examples
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