I was reading Code by Charles Petzold and i found myself struggling with the rules of sets.

On the 81th page it says :

The commutative, associative, and distributive rules all hold for Boolean algebra. What's more, in Boolean algebra the + operator is distributive over the x operator. This isn't true of conventional algebra:

W + (B x F) = (W + B) x (W + F)

The union of white cats and black female cats is the same as the intersection of two unions: the union of white cats and black cats, and the union of white cats and female cats. This is somewhat difficult to grasp, but it works.

So i used Venn Diagrams to demonstrate my supposition for the first part of equation Click here

And the second Click

My question is what does the C (filled blue) set contain? I hope that i filled that right. I personally think , that ALL white cats and also Cats that are female and black at the same time I got it by the 1st diagram, but not the second. I just can't understand what the word "OR" means in this context. Please leave feedback on my 2nd diagram and correct if i am mistaken. Thanks in advance.

  • $\begingroup$ "OR" is set union: "Cats are either male or female" means that the set of all Cats is "made of" the set of male Cats "plus" (set union) the set of female Cats. $\endgroup$ – Mauro ALLEGRANZA Apr 2 '17 at 11:43
  • $\begingroup$ "AND" is set intersection : male Cats are both male and cats, i.e. they are the intersection of the set of all Cats with the set of all male "animals". $\endgroup$ – Mauro ALLEGRANZA Apr 2 '17 at 11:44
  • $\begingroup$ Your first Venn diagram must be redraw: W + (B x F) does not "cover " all Cats. We have also Black male Cats (that are neither White nor Black female). $\endgroup$ – Mauro ALLEGRANZA Apr 2 '17 at 11:47


When using Venn diagrams you want each circle to represent a 'simple' or 'atomic' class of objects. So for example: 'cats' would be a simple category. Or: 'white things'. But you really don't want to use 'things that are cat or black'. Indeed, even 'white cats' is already a combination of 'things that are cat and that are white': you really want to leave out logical operations in your definition for a single circle. Indeed, you use intersection, union, etc. to think about 'and', 'or', etc.

So in your case: have circles for 'cats', 'white things' and 'black things', and try again.


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