E1.2 - Three sets and set relationships

Membership, subsets and the empty set

Is “$3$ belongs to $A$” the same kind of statement as “every element of $B$ belongs to $A$”? The first talks about one object; the second compares two complete collections. Their symbols need to preserve that distinction.

Let $A=\{1,2,3,4\}$ and $B=\{2,4\}$. The number $2$ is an element of $A$. Both elements of $B$ are in $A$, so the entire set $B$ fits inside $A$.

To disprove a subset claim, one counterexample suffices. For $D=\{2,5\}$, the element $5$ shows $D\nsubseteq A$; the fact that $2$ is shared cannot rescue the claim. To prove a finite subset claim, check every element of the proposed subset.

[DIAGRAM: asset_name: Subsets and disjoint sets; asset_slug: extended_sets_relationships; caption: Containment and separation represent different relationships.; description: Two panels show universal rectangles. On the left, circle B lies entirely inside larger circle A, with elements 2 and 4 in B and 1 and 3 in A outside B. On the right, separate circles P and Q contain 1 and 3, and 2 and 4 respectively, with no overlap.]

The right-hand sets are disjoint: they have no common elements. Their intersection is the empty set, written $\varnothing$, so $P\cap Q=\varnothing$ and $n(P\cap Q)=0$. The set $\{0\}$ is different: it contains one element, zero. Empty braces $\{\}$ also represent the empty set.

Every set is a subset of itself. The empty set is a subset of every set, because it has no element that could violate the condition “every element is in the other set”. That does not make it an element of every set: $\varnothing\notin\{1,2\}$, since the two elements are the numbers $1$ and $2$.

Check the kind of object on each side of the symbol in this question.

Three-set Venn regions

Three circles arranged with every kind of overlap produce eight separate regions, including the region outside all three. An element in all three belongs in the central overlap once. Begin there when totals include it, then move out to regions in exactly two sets and finally regions in only one.

Suppose a group has $40$ students. Sets $A,B,C$ contain $20,18,17$ students respectively. The counts in the pairwise intersections are $n(A\cap B)=8$, $n(A\cap C)=7$ and $n(B\cap C)=6$, and $3$ students belong to all three. Each given pairwise count includes the central three.

[DIAGRAM: asset_name: Eight regions of a three-set Venn diagram; asset_slug: extended_sets_three_regions; caption: Pairwise intersections include the central region unless “only” excludes it.; description: Three overlapping circles A, B and C lie in a universal rectangle of 40 students. Region counts are A only 8, B only 7, C only 7, A and B only 5, A and C only 4, B and C only 3, all three 3, and outside all three 3.]

Keep all counts and change the target to students outside every set. This requires the remaining regions, each counted once.