* Note: This is my understanding of some idea in Coursera Course Intro to Mathematical Thinking. Initially I drew these pics just for myself, but someone in discussion forum ask for a clear explanation, and the forum is not embedded with good markdown editor. So I put it here for hyper link reference.
* Note2: For better user experience, you may duplicate the window, one for showing the pics, the other for reading the text.

Two weird imply statement

The six statements all mean '\phi implies \psi',
four on the left seems quite reasonable:

• If \phi, then \psi
• \phi is sufficient for \psi
• \psi if \phi
• \psi whenever \phi

But the rest two on the right may be quite counterintuitive:

• \phi only if \psi
• \psi is necessary for \phi

Here are two responding example to help you get the idea:

• 'Attend Tour de France' only if 'can ride a bike'.
• 'Can ride a bike' is necessary for 'attend Tour de France'.

They both mean the same thing: If 'attend Tour de France' happen, 'can ride a bike must happen'.

Apart from rely on intuitive example, is there some more universal thinking approach that help us digest the idea of 'sufficient and necessity' well?

Here's how I take it.

Sufficiency

(Don't worry about the handwriting, I've put text below the pic.)

• 1. \phi is loosely attached to several buttons(or light bulbs if you like), one of them is called \psi.
• 2. When you light up \phi, some arbitrary button will shine.
• 3. No matter which set will shine, you find out that \psi will always be one of it.
• 4. So the relationship between \phi and \psi is bound tight.
• 5. When \phi happen, \psi is definitely going to happen. So we say '\phi is sufficient for \psi'.

The idea of 'sufficiency' is quite straightforward, however when we look at necessity, it's a slightly more complicated story.

Necessity

• 6. Now \phi and \psi happily live ever after...Wait, that's not the end of the story. ( Notice the 'time flow' arrow below the pic)
• 7. Going backwards alongside the time flow, there's some other buttons that will light \phi up.
• 8. It can be the upper one, the middle one, and of course the lower one. None of them can be called 'being necessary for \phi', because there are three of them. For each one, when it don't roll up the sleeves, some other guy can light up \phi. There's no necessity.

• 9. There's no one necessary for \phi until someone took part in...Let's name it 'GUY'.
• 10. Now this 'GUY' can be called 'necessary for \phi', because it monopolies all the roads lead to \phi, it's a must-go-through.
• 11. If \phi is lit up, no matter what, this 'GUY' was lit up beforehand.
• 12. BTW, 'GUY' might have some subsidiaries other than \phi. 13) So, when 'GUY' lights up, it's quite arbitrary whether \phi will light up.
• 14. On the contrary, whenever \phi lights up, 'GUY' must light up, since it's the only access to \phi from left to right.

• 15. When 'GUY' happen to be \psi, 16) this relationship can be simplified to :
      \psi <==== \phi ====> \psi
      ( Of all the connection, only these two strong arrow are 100% bound. )
• 17. '\phi is sufficient for \psi':
      \phi ====> \psi
      When \phi lights up, \psi lights up.
• 18. '\psi is necessary for \phi':
      \psi ====> \phi
      When \phi lights up, \psi must have been lighten up, because there's on other way to get \phi lit.
• 19. (17) and (18) basically say the same thing: ** * \phi implies \psi * **, but in two different angles: ** sufficiency ** and ** necessity **.