4 Attachment(s)

Set Intersection Ratio and Set Difference Inequality Proof

The problem in question is Given that A_{1} and A_{2 }are subsets of a universal Set and that

Attachment 24287

Prove also that

Attachment 24286.

I've tried a number of things, such as multiplying similar ratios and forms of the number one, however it always leaves me with an expression that doesn't simplify. I've also tried to use a number of the basic identities of union, intersection, set difference, and complements, but to no avail.

I'm avoiding using the base formula of the principle of inclusion/exclusion, as it is not introduced until after this problem, however I'm not opposed to using it if it is necessary. I'm mostly looking for ideas for what I should try (as I have run out by now), rather than an overall solution, but I'd appreciate any input.

Thank you.