Any connection between pet and petty?

Hi, is there any connection between pet and petty? I mean, a pet is a small animal and petty describes something insignificant or little too. Or is it a coincidence that they both have one syllable in common?

Thanks in advance,

TOEIC short conversations: A customer is getting put through to his accountant.[YSaerTTEW443543]

There might well be a connection between both terms!

According to an accepted theory, pet derives from ‘pety’ (small) in an old Scottish dialect. Petty comes from old French and is a variant of ‘petit’ (small, little).

Now, does that mean that, if your pet is an elephant, you cannot call it a pet?