Proving inverse implication by conversion

Question

I have proven logically that the inverse of an implication is true if and only if the converse of said implication is true (as shown below).

proposition 1: k has same parity as 2j

proposition 2: k is even

implication: If k is even, then k has the same parity as 2j.

inverse: If k does not have the same parity as 2j, then k is not even.

converse: If k is not even, then k does not have the same parity as 2j.

My question is, how can I write this in a mathematical notation? I know that I can write the propositions' mathematical notations as such:

k => 2j

But how should I write my implication? Thanks in advance!

The converse of the implication should be
"If $k$ is even, then $k$ has the same parity as $2j$."

What you stated is the contrapositive. — user67953840, Jan 23 '15 at 17:17
@MarkLao thanks, but isn't your correction just dealing with semantics? Couldn't it just swap proposition 1 and 2 to correct my implication? — Jim22150, Jan 23 '15 at 17:21
If you swap propositions 1 and 2, what you stated as converse becomes the inverse. — user67953840, Jan 23 '15 at 17:23
Just to be clear, the converse you stated is not the converse of the implication. But it is the converse of the inverse of the implication that you stated. — user67953840, Jan 23 '15 at 17:26
Okay, I fixed the mistake. Could you please provide any thought on the mathematical notation of the proposition? Thanks again. — Jim22150, Jan 23 '15 at 17:26
I actually don't understand what you want to do. By "write in a mathematical notation", do you mean to write mainly using math symbols instead of words? — user67953840, Jan 23 '15 at 17:28
@marklao Yes, how can I write the above if then else statement mathematically? — Jim22150, Jan 23 '15 at 17:35

score 0 · Accepted Answer · edited Jun 12 '20 at 10:38

0

Assuming to "write this in a mathematical notation" means to write mainly using math symbols instead of words, then you may say:

Let $E$ be the set of even integers.

If $k \in E$, then $k \equiv 2j\pmod2.$

Or the more extreme version:

$k \in E \Rightarrow k \equiv 2j\pmod2$.

But the way you stated the implication is perfectly fine, if not better.

edited Jun 12 '20 at 10:38

Community

answered Jan 23 '15 at 17:39

user67953840

1 Answers1