When do we say converse of something in Proving?Its Simple!Please Help?

Question

Well, I didnt pay attention for this one...

But when we say a statement and we are going to write a reason for it like this:

Triangle ABC is isoceles l Converse of ITT

When do we say like converse of ITT or just ITT?

Thank you!

Answer 1

A converse is the reversal of the statements in a proposition.

The converse is not necessarily true.

If A, then B.
(this is called an implication)

converse =
If B then A.

A = it rains
B = I carry an umbrella

If A, then B
(For this discussion let us accept that this statement is always true):
If it rains, then I carry an umbrella.

However, I could be carrying an umbrella for other reasons, so that the converse:

If I carry an umbrella, then it rains
(if B, then A)
is not necessarily true.

I could be carrying a new umbrella I just purchased as a gift for somebody else.

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The contrapositive is different.

If A, then B

contrapositive:

If not B, then not A.

If I am not carrying an umbrella, then it is not raining. (Otherwise, this would imply that there could be rain without me carrying an umbrella, contrary to the initial implication).

If the statement is true, then the contrapositive must also be true.

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Might as well complete with the fourth type of statement: the inverse:

if not A, then not B.

If it does not rain, then I do not carry an umbrella.

The truth of the initial implication does not guarantee the truth of the inverse (use the same example as earlier, with the gift-umbrella).

However, if you are given that both the implication AND its inverse are true, then, you have:

- an "if-and-only-if" situation (for example: if it rains, and only if it rains, I carry an umbrella); and
- the converse is true: (if I carry an umbrella, then it rains); and
- the contrapositive remains true.