# Laws of set algebra

by sara_87
Tags: algebra, laws
 P: 773 Question: Use the laws of set algebra to show that for the sets X,Y,Z: a) X + X’Y + ZX’ = X + Y + Z b) XY + X’Y’ + X’Y + XY’ = U c) Z’(X + Y’)(X’ +Z) = (X + Y’ + Z)’ d) XYZ + X’ + Y’ + Z’ = U Answer: I can prove these using venn diagrams but what are the laws of set algebra? also U means the union of X, Y and Z...no?
 Emeritus Sci Advisor PF Gold P: 16,091 I would imagine things like the distributive laws, the associative laws, DeMorgan's laws, et cetera.
Mentor
P: 8,325
 Quote by sara_87 also U means the union of X, Y and Z...no?
U generally denotes the universal set; a set large enough to contain all sets under consideration (for the given question). For example, Ac=U\A.

(Another way to think of it, is as the "rectangle" in which the sets are drawn when constructing a Venn diagram) In this instance, it is enough to take U to be the union of X, Y and Z.

HW Helper
P: 9,396
Laws of set algebra

 Quote by sara_87 I can prove these using venn diagrams but what are the laws of set algebra?
No, you can't prove them using venn diagrams. You can justify them, but I doubt anyone would accept Venn diagrams as a 'proper' proof. It is easy to turn Venn diagram arguments into proper arguments though.
 P: 773 'It is easy to turn Venn diagram arguments into proper arguments though.' yeah?...how? could you give me an example?
 Sci Advisor HW Helper P: 9,396 Look at the diagram. A set equality A=B is always best shown (if no clever methods are at hand) by demonstrating that anything in A is in B, and vice versa. If you look at the venn diagram for a), say, you can read off how to do this because you can see where elements in subsets of the LHS can be found in subset of the RHS. Of course, here you just use the laws set algebras some of which are (and these are comutative operations) XX=X X+X=X X'=U\X (set difference), or U=X+X' XX'=0 UX=X U+X=U In particular Y=XY+X'Y
Math
Emeritus