biconditional or double implication

Double implication.
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In my last video 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 video. If you not seen my last video on conditional search my channel by typing the word “mathstips4u” in the search bar of YouTube or Dailymotion or Google search bar or you may visit my blog https://mathstips4u.blogspot.com. You will get list of all my previous videos or posts. Select the video you want study try to solve all the exercise in it and verify your answers with my next video. If you have any doubt, please comment me in the comment section. I will try my level best to satisfy you.
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 video 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 〖 x〗^2 = x.
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 = x, then x = 1.
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 video we are going to learn Statement pattern and Logical Equivalence and some of their examples
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