Implication or conditional in Mathematical Logic
Hello friends, Welcome to my blog mathstips4u.
In this blog I posted articles on Mathematics for
the students of 11th and 12th class. In these posts I
have given tips and tricks that helps the students to overcome fear of facing
board exam.
In my last post we have seen negation and its truth
table.
In this post we are going to learn implication (or
conditional) and its truth tables and some of its examples.
So we shall begin with
Implication or conditional [→]:
Let p and q be two simple
statements. Then the compound statement ‘If p then q’ is called the conditional
or implication, and denoted by p → q or p => q. It is read as p implies q.
In the implication p → q, p is
called antecedent or hypothesis
and q is called the consequent or conclusion.
p → q is defined to
have the truth value ‘false’ if p has truth value ‘true’ and q has truth value
‘false’. In all other cases, p → q is defined to have the truth value ‘true’.
Truth table of implication p → q
p
|
q
|
p → q
|
T
|
T
|
T
|
T
|
F
|
F
|
F
|
T
|
T
|
F
|
F
|
T
|
Remember: p → q and q → p both are not the same.
Following are phrases in
English which are equivalent to (p → q)
1.p is sufficient
for q.
2. q is
necessary for p.
3. p only if q.
4. q follows p.
5.q provided p.
6. q if p.
Ex. Express following in
symbolic form.
1. Visiting Himalayas implies peace of mind.
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
= x.
6. The sufficient condition
for being rich is to be rational.
7. Getting bonus is
necessary condition for me to purchase a car.
Solution:
1. Visiting Himalayas implies peace of mind.
First we write the given statement by using if …then.
‘If I visit Himalayas, then I
will get peace of mind’
p: I visit Himalayas.
q: I get peace of mind.
The symbolic form is p → q.
The remaining are left intentionally as
exercise for you. Try it or wait till my next post on bi-conditional is uploaded
in my blog “mathstips4u”.
In this way we have seen
implication and its truth tables and some of its examples.
In my next video we are going
to learn Bi-conditional or double implication and its truth table.
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