The universal set U contains every element under discussion, and every set is a subset of it. A Venn diagram draws U as a rectangle and each set as a closed region, so that set relations become regions.
In any problem about sets, the elements are taken from some larger collection. For sets of whole numbers, that might be the natural numbers. For sets of students, it might be all the students in a school.
This collection is the universal set, written U. Every set in the problem is a subset of U.
U must be fixed before statements such as “the elements not in A” have a definite meaning. “Not in A” means “in U but not in A”, so it depends on U.
E is the set of even numbers. What is “everything not in E”, and why does the answer depend on U?
If U is the set of natural numbers, the elements not in E are the odd natural numbers: 1, 3, 5, …
If U is the set of real numbers, the elements not in E include the odd integers, the negative non-even integers, every fraction that is not an integer, and every irrational number.
The same set E gives two different answers. The answer is determined by U.
A Venn diagram represents sets as regions of a plane. U is drawn as a rectangle. Each set is drawn as a closed curve, usually a circle, inside the rectangle. An element of a set is placed inside that set’s curve; an element not in the set is placed outside it.
Two overlapping circles A and B divide the rectangle into four regions: elements in A only, in B only, in both A and B, and in neither. Each element of U lies in exactly one of these regions, determined by which of A and B it belongs to.
Four regions from two sets. The rectangle is U. The circles A and B divide it into four regions, and every element of U lies in exactly one.
A Venn diagram replaces a statement about membership with a region that can be pointed to. “Elements in both A and B” is the overlap. “Elements in A but not B” is the part of A outside the overlap.
Each operation in the next session corresponds to one region or a combination of regions. A diagram is a picture of a particular case, so it is used to understand and check a statement. A proof of a general statement still uses the definitions.
In a two-set Venn diagram, which region contains the elements that are in A but not in B?
The part of circle A that lies outside the overlap.
Circle A contains every element of A. The overlap contains the elements of A that are also in B. Removing the overlap from A leaves the elements of A that are not in B.
This region is the set difference A − B, defined in the next session.
1The set that contains every element under discussion, of which all other sets are subsets, is the ____ set, written U.
2In a Venn diagram, the universal set is drawn as a…
3Why must U be fixed before describing “the elements not in A”?
4Two overlapping circles in a Venn diagram divide U into how many regions, including the region outside both?
5The overlap of circles A and B represents the elements that are…
6The elements in A but not in B lie…
Next: union, intersection, difference and complement, each shown as a region.
Noescio