=Paper=
{{Paper
|id=Vol-2116/paper5
|storemode=property
|title=Visualization of Set Inclusion with Gloves
|pdfUrl=https://ceur-ws.org/Vol-2116/paper5.pdf
|volume=Vol-2116
|authors=Toshio Suzuki
|dblpUrl=https://dblp.org/rec/conf/diagrams/Suzuki18
}}
==Visualization of Set Inclusion with Gloves==
https://ceur-ws.org/Vol-2116/paper5.pdf
Visualization of Set Inclusion with Gloves
Toshio Suzuki⋆
Department of Mathematical Sciences, Tokyo Metropolitan University,
Minami-Ohsawa, Hachioji, Tokyo 192-0397, Japan
toshio-suzuki@tmu.ac.jp
Abstract. For educational purpose, we introduce visualization media
for implication in classical propositional logic and inclusion of finite sets.
On the former, we propose a pictorial truth table by means of a pair of
pot holders. A mitten of open (closed, respectively) state denotes a true
(false) proposition. We let the left mitten dive under the right mitten.
Then the left mitten totally hides behind the right mitten exactly when
“the left proposition implies the right proposition.” On the latter, we
represent a set by a glove, where each finger denotes whether an element
belongs to the set. The left glove totally hides behind the right glove
exactly when the left set is a subset of the right set. A slightly more
abstract diagram for set inclusion is also introduced.
Keywords: novel forms of set visualization; mathematics education; im-
plication; set inclusion
2010 MSC: 00A66, 97E30, 97E60
1 Introduction
In classical propositional logic, for propositions p and q, the truth table of “p
implies q” (in symbols, p → q) is given by Table 1. For example, refer to [1,
Chapter 0]. Here, T denotes true and F denotes false. In other words, p → q has
the same truth table as “(not p) or q”.
p q p→q
T T T
T F F
F T T
F F T
Table 1. truth table of implication
⋆
This work was partially supported by Japan Society for the Promotion of Science
(JSPS) KAKENHI (C) 16K05255.
Y.Sato and Z.Shams (Eds.), SetVR 2018, pp. 68-75, 2018.
Visualization of Set Inclusion with Gloves Suzuki
Given sets A and B, set inclusion is defined as follows. A is a subset of B (in
symbols, A ⊆ B 1 ) if the following holds.
For all x (x ∈ A → x ∈ B)
Here, “→” is the implication of classical propositional logic.
We are interested in visualization suitable for beginner course of sets and
logic. In this paper, on the implication of classical propositional logic and finite
set inclusion, we are going to propose new such visualization. Our visualization
media are completely different from those of Venn [4]. Instead of circles overlap-
ping each other, we use gloves with wire in them. The left glove totally hides
behind the right glove exactly when the left set is a subset of the right set.
In section 2, we give a pictorial truth table of implication. In section 3, we
give pictorial expressions of finite set inclusion. In section 4, we introduce a
slightly more abstract diagram for set inclusion. The ideas of visualization in
this paper are developments of those in the author’s previous books [2, 3].
2 Pictorial truth table of implication
In this section, for the implication of classical propositional logic, we introduce
a pictorial truth table. We prepare a pair of rectangular pot holders with wire
in them. For simplicity, we call them mittens. We assume that a mitten has two
possible states, open state and closed state. Fig. 1 is a mitten of open state on a
table, where its palm is upward. By bending the perm of the mitten inside, we
get a mitten of closed state (Fig. 2).
Fig. 1. open state Fig. 2. closed state
Suppose that ℓ and r are propositions. Table 2 is a our pictorial truth table
of “ℓ implies r”. The table is similar to Table 1 except that pictures are inserted
into cells. In the column ℓ, we append a picture of open upward left palm (of
a mitten, and so forth) to T, and append a picture of closed left upward palm
to F. In the column r, we append a picture of open downward right palm to T,
and append a picture of closed downward right palm to F. Now, at each row, we
1
In some literatures, A ⊂ B is used.
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Visualization of Set Inclusion with Gloves Suzuki
let the left mitten dive under the right mitten. In the new column “overlay”, we
put a resulting picture. The column of ℓ → r is the same as before.
ℓ r overlay ℓ→r
T
T T
F
T F
T
F T
T
F F
Table 2. pictorial truth table of implication
Then, open upward left palm totally hides behind open downward right palm
exactly like total solar eclipse. Open upward left palm does not totally hide be-
hind closed downward right palm. Closed upward left palm totally hides behind
open downward right palm. Closed upward left palm totally hides behind closed
downward right palm. To sum up, in the columns overlay and ℓ → r, “totally
hides behind” agrees with the truth of ℓ → r.
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Visualization of Set Inclusion with Gloves Suzuki
3 Visualization of finite set inclusion
In this section, we are going to visualize finite set inclusion. In the case where
our universal set is a singleton, the visualization is achieved as a special case
of Table 2. Suppose that E is a singleton whose unique element is 1. In other
words, E = {1}. In this case, E has exactly two subsets. One is E, and the other
is ∅ (the empty set).
Suppose that L and R are subsets of E. Let ℓ be proposition “1 ∈ L” and r
be proposition “1 ∈ R”. Then, open state denotes E, and closed state denotes ∅.
In other words, the four possible truth values of (ℓ, r); (T, T ), (T, F ), (F, T ) and
(F, F ) mean the four possible values of (L, R); (E, E), (E, ∅), (∅, E) and (∅, ∅),
respectively. The proposition “ℓ → r” agrees with “L ⊆ R”. Thus, “totally hides
behind” agrees with “is a subset of”. Therefore, Table 2 visualizes set inclusion
in this case.
Next, we are going to investigate the case where our universal set E is
{1, 2, 3, 4, 5}. One way to visualize this situation is to use five pairs of mittens.
Instead, we use a pair of gloves. Each finger of a glove is an alternative of a mit-
ten. Each finger has two possible states, stretched state and bent state. Suppose
i is one of 1,2,3,4 and 5. Stretched state of finger i denotes proposition “i belongs
to the set denoted by the glove”. Bent state of finger i denotes proposition “i
does not belong to the set denoted by the glove”.
We put the left palm upward and the right palm downward. In the case where
all the fingers are stretched in the both gloves, the set L denoted by the left glove
is {1, 2, 3, 4, 5}, and the set R denoted by the right glove is the same set (Fig. 3).
We let the left glove dive under the right glove. The left glove of Fig. 3 totally
hides behind the right glove of Fig. 3. In this case, {1, 2, 3, 4, 5} is a subset of
{1, 2, 3, 4, 5} (Fig. 4).
Fig. 3. L = {1, 2, 3, 4, 5}, R = {1, 2, 3, 4, 5} Fig. 4. L ⊆ R
We do not observe all of the 1024 possible combinations. Instead we show
some examples to illustrate our idea. We bend the 4th finger and 5th finger of
the left glove. The resulting new L is {1, 2, 3}, and R is {1, 2, 3, 4, 5} (Fig. 5). We
let the left glove dive under the right glove. The left glove of Fig. 5 totally hides
behind the right glove of Fig. 5. {1, 2, 3} is a subset of {1, 2, 3, 4, 5} (Fig. 6).
71
Visualization of Set Inclusion with Gloves Suzuki
Fig. 5. L = {1, 2, 3}, R = {1, 2, 3, 4, 5} Fig. 6. L ⊆ R
For the next example, we bend the 1st, 4th and 5th finger of the right glove.
Thus L = {1, 2, 3} and R = {2, 3} (Fig. 7). The left glove does not totally hide
behind the right glove. {1, 2, 3} is not a subset of {2, 3} (Fig. 8).
Fig. 7. L = {1, 2, 3}, R = {2, 3} Fig. 8. L ̸⊆ R
In the case where L = ∅ and R = {2, 3} (Fig. 9), the left glove hides behind
the right glove. ∅ is a subset of {2, 3} (Fig. 10).
Fig. 9. L = ∅, R = {2, 3} Fig. 10. L ⊆ R
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Visualization of Set Inclusion with Gloves Suzuki
In the case where our universal set is {1, 2, . . . , 10}, we can carry out the
same thing as above. We use two pairs of gloves, say pair A and pair B (Fig. 11).
We let the left glove of pair A dive under the right glove of pair A, and let the
left glove of pair B dive under the right glove of pair B (Fig. 12).
Fig. 11. L = {1, 2, · · · , 10}, R = {1, 2, · · · , 10}
Fig. 12. L ⊆ R
4 Discussion
The following are examples of cases where we need extra caution.
– The case where the cardinality of the universal set is not a multiple of 5.
– The case where elements of the universal set is not a number, say fruits.
– The case where the universal set is an infinite set.
In order to adapt to these cases, we introduce a slightly more abstract visu-
alization. We draw a baseline. Under the baseline, we write labels for elements,
for example 1,2,3,4 and 5. A square above label n denotes an upright nth finger.
If there is only the baseline above label n then this denotes a bent nth finger.
For example, counterpart to Fig. 7 is given by Fig. 13.
1 2 3 4 5 1 2 3 4 5
Fig. 13. L = {1, 2, 3}, R = {2, 3}
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Visualization of Set Inclusion with Gloves Suzuki
In Fig. 13, we may take the universal set as {1, 2, 3}. It is no longer unnatural
to shorten the baseline and erase labels 4 and 5. Fig. 14 denotes the same sets
{1, 2, 3} and {2, 3} as Fig. 13. The only difference is what set is considered as
the universal set.
1 2 3 1 2 3
Fig. 14. L = {1, 2, 3}, R = {2, 3}
In the case where L = { apple, lemon, orange } and R = { lemon, orange },
we replace labels 1,2 and 3 by apple, lemon and orange respectively (Fig. 15).
Another order, for example lemon, orage, apple is possible, provided that the
same order of labels are adopted in the two sets.
ap
lem
or
ap
lem
or
an
pl
a
pl
ng
on
on
e
ge
e
e
Fig. 15. L = { apple, lemon, orange }, R = { lemon, orange }
Next, we observe the cases of inifinite sets. For transfinite ordinals and axiom
of choice, consult an introductory textbook on set theory [1].
We look at the case where the universal set is von Neumann ordinal ω + 4 =
{0, 1, 2, . . . , ω, ω + 1, ω + 2, ω + 3} and a subset L = {1, 3, . . . , 2n + 1, . . . , ω +
1, ω + 3} is given. Then under the baseline, we write elements of the universal
set in order from the smallest. In Fig. 16, we draw L only. If another subset R
is given, we draw a similar diagram for R.
0 1 . . . 2n 2n+1 . . . ω ω+1 ω+2 ω+3
Fig. 16. {1, 3, . . . , 2n + 1, . . . , ω + 1, ω + 3}
In general, if the universal set is a von Neumann ordinal, we may imagine a
diagram similar to Fig. 16 in our mind. Under axiom of choice, for any set X,
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Visualization of Set Inclusion with Gloves Suzuki
there is a bijection from X to an ordinal, thus if once we fix such a bijection we
may imagine a similar diagram for X. However, the general case of an infinite
set X is not straightforward yet.
5 Concluding remarks
Beginners of sets and logic sometimes confuse exact definition of “implies” with
meaning that the phrase “if · · · then” have in everyday conversation. “F implies
T” would sound strange for them. Similar confusion would occur between the
technical term “is a subset of” and the phrase “is included by”. “The empty set
is a subset of a given set” would sound strange for beginners.
Advanced learners can understand the truth table (Table 1) by means of the
Boolean algebra of exactly two elements. Nevertheless, such understanding is
appropriate only for those who are already familiar with sets and logic. Thus,
visualization is valuable for beginners of sets and logic.
In our pictorial truth table of implication (Table 2), “totally hides behind”
coincides with the truth of implication. By substituting gloves for mittens, we
visualized finite set inclusion. We visualized the empty set by a closed glove.
As a by-product of our visualization media, we can visualize difference be-
tween “is an element of” with “is a subset of”. Number i is an element of a
given set exactly in the case where finger i is stretched in the glove expressing
the set. On the other hand, set L is a subset of set R exactly in the case where
the upward left palm expressing L totally hides behind the downward right palm
expressing R.
In order to see for which type of learners our approach is effective, feedback
from various learners and teachers would be helpful.
Acknowledgments
We are grateful to anonymous reviewers for helpful comments.
References
1. Kenneth Kunen. The foundations of mathematics, College Publications, London,
2009.
2. Toshio Suzuki. Logic literacy, Baifukan, Tokyo, 2009. (written in Japanese. ISBN:
978-4563003838)
3. Toshio Suzuki. An introduction to sets and logic with exercises, Morikita, Tokyo,
2016. (written in Japanese. ISBN: 978-4627061910)
4. John Venn. Symbolic logic, second edition, MacMillan, London, 1894. Reprint:
Chelsea, New York, 1971.
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