Welcome to a lesson
on converse and contrapositive of an implication.
Remember, an implication or conditional
is a molecular statement of the form, if P then Q
where P and Q are statements.
The converse is obtained
by swapping the conclusion and hypothesis,
meaning, swapping Q and P,
the converse of if P then Q, is if Q then P.
The converse is not logically equivalent
to the original implication.
That is, whether the converse of an implication is true
is independent of the truth of the implication.
The contrapositive is obtained by implying that the negative
or negation of the conclusion implies the negative
or negation of the hypothesis.
The contrapositive is if not Q, then not P.
An implication in its contrapositive
or logically equivalent, meaning,
they are either both true or both false.
This lesson won't focus on negation,
but the negation of an implication is obtained
by taking the conjunction, meaning and,
of the hypothesis and the negation of the conclusion.
The negation of if P then Q
is the equivalent to P and not Q.
Mathematics is overflowing with examples
of true implications, which have a false converse.
Here are two examples.
The implication, if a number greater than two is prime,
then the number is odd is true.
However, the converse is false.
Just because a number is odd does not mean that it is prime.
Similarly, if a shape is a square,
then it is a rectangle is a true implication.
But again, the converse is false.
If a shape is a rectangle, then it is a square is false.
However, sometimes, the converse
of a true statement is also true.
For example, the Pythagorean theorem has a true converse.
Beginning with the implication,
if a triangle with sides, A, B, and C,
with C being the longest side, is a right triangle,
then A squared plus B squared equals C squared.
This is a true statement.
The converse of this implication
is if a squared plus B squared equals C squared,
then the triangle with sides A, B, and C
is a right triangle is also true.
Whenever you encounter an implication in mathematics,
it is always reasonable to ask whether the converse is true.
The contrapositive, on the other hand,
always has the same truth value as its original implication.
This can be very helpful in deciding
whether an implication is true,
often it is easier to analyze the contrapositive.
The contrapositive of the original implication
involving the Pythagorean theorem is,
if A squared plus B squared doesn't equal C squared,
then the triangle with sides, A, B, and C
is not a right triangle.
And because the original implication is true,
we know this contrapositive is also true.
Let's take a look at one more example.
We want you to determine whether the given implication
is true or false as well as its converse.
And the implication is, if you draw any nine playing cards
from a regular deck, then you'll have at least three cards
all of the same suit.
I've included a picture of a deck of cards below
to help analyze the implication.
And this statement is true.
However, the original implication
is a little hard to analyze
because there are so many different combinations
of nine cards, but let's consider the contrapositive.
The contrapositive of the original implication is,
if you don't have at least three cards
all of the same suit, then you don't have nine cards.
It may be easier to see why this is true.
You can at most have two cards of each of the four suits
for a total of eight cards or fewer.
Meaning, you can have at most two cards
of each of the four suits
for a total of eight cards or fewer.
For example, you could have two hearts,
two clubs, two diamonds, and two spades.
But as soon as you pick that ninth card,
you are going to have at least three cards
all of the same suit.
And now, let's consider the converse
of the original implication.
So remember, to form the converse,
we interchange the conclusion in hypothesis or Q and P.
The converse is,
if you have at least three cards all the same suit,
then you have nine cards.
And the converse is false.
You could have three spades and nothing else.
Note that to demonstrate that the converse,
which is an implication is false,
we provided an example where the hypothesis is true,
meaning, we have three cards of the same suit,
but the conclusion is false,
meaning, we do not have nine cards.
I hope you found this helpful.