Alan

In your diagram, A represents dog lovers and B represents cat lovers.

A' means everything that is not in A. 

The inverted U means intersection.

Hence \(A'\cap B\) means everything that is both not in A  and is in B simultaneously.  That is indicated by the shaded region in your diagram.

As for whether the numbers are correct, that will depend on the data specified for your problem.  Your data suggests there are 8 people in total, 4 of whom love dogs but not cats, one who loves cats but not dogs and 3 who love both dogs and cats. 

We would need to see your Venn diagram to make any sense of it.  In general, any part(s) of a Venn diagram could be shaded - the shading is just to distinguish it from some other region, or to draw your attention to a particular region.

Upload your Venn diagram.

This question keeps coming up!  See my solution at https://web2.0calc.com/questions/help-please_7362#r4 

(No need for an approximate numerical method).

"Among all pairs of numbers (x, y) such that 3x + y = 16, find the pair for which the sum of squares, x² + y², is minimum. Write your answers as fractions reduced to lowest terms."

Here's the answer for part (a):

See if you can use the same technique for part (b).

Looks like there are 19 possible finite roots!!  Use a numerical technique like Newton-raphson to find them:

This relates to the Collatz conjecture.  For the specific numbers quoted there are 15 different integers involved (that is, if I've interpreted the question correctly!), namely: 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1

See https://web2.0calc.com/questions/what-is-the-polar-form-of-the-equation-of-x-6-2-y-2-36#r2 

As follows:

Let speed of slower point = v m/s.  Distance it travels in 4 seconds: s1 = 4v m

Distance travelled by faster point in 4 seconds:  s2 = 4(v + 5) m

s1 + s2 = 100 m,  so  4v + 4(v + 5) = 100,  or v = 10m/s.  faster point speed = 15m/s