diagram for set inclusion is also introduced.
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 T T T F F T F F p ! q
T F T
T ⋆ This work was partially supported by Japan Society for the Promotion of Science (JSPS) KAKENHI (C) 16K05255.
Given sets A and B, set inclusion is de ned as follows. A is a subset of B (in symbols, A B1) if the following holds.
For all x (x 2 A ! x 2 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 nite
set inclusion, we are going to propose new such visualization. Our visualization
media are completely different from those of Venn [
In section 2, we give a pictorial truth table of implication. In section 3, we
give pictorial expressions of nite 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 [
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).
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. 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 T T F F r T F T F overlay ℓ ! r
T F T T
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 behind 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. 3
In this section, we are going to visualize nite 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 = f1g. 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 2 L" and r be proposition \1 2 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 f1; 2; 3; 4; 5g. One way to visualize this situation is to use ve pairs of mittens. Instead, we use a pair of gloves. Each nger of a glove is an alternative of a mitten. Each nger has two possible states, stretched state and bent state. Suppose i is one of 1,2,3,4 and 5. Stretched state of nger i denotes proposition \i belongs to the set denoted by the glove". Bent state of nger 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 ngers are stretched in the both gloves, the set L denoted by the left glove is f1; 2; 3; 4; 5g, 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, f1; 2; 3; 4; 5g is a subset of f1; 2; 3; 4; 5g (Fig. 4).
We do not observe all of the 1024 possible combinations. Instead we show some examples to illustrate our idea. We bend the 4th nger and 5th nger of the left glove. The resulting new L is f1; 2; 3g, and R is f1; 2; 3; 4; 5g (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. f1; 2; 3g is a subset of f1; 2; 3; 4; 5g (Fig. 6).
For the next example, we bend the 1st, 4th and 5th nger of the right glove. Thus L = f1; 2; 3g and R = f2; 3g (Fig. 7). The left glove does not totally hide behind the right glove. f1; 2; 3g is not a subset of f2; 3g (Fig. 8).
In the case where L = ∅ and R = f2; 3g (Fig. 9), the left glove hides behind the right glove. ∅ is a subset of f2; 3g (Fig. 10).
In the case where our universal set is f1; 2; : : : ; 10g, 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). { 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 in nite set.
In order to adapt to these cases, we introduce a slightly more abstract visualization. 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 nger. If there is only the baseline above label n then this denotes a bent nth nger.
For example, counterpart to Fig. 7 is given by Fig. 13.
1 2 3 4 5 1 2
3 4 5
In Fig. 13, we may take the universal set as f1; 2; 3g. It is no longer unnatural to shorten the baseline and erase labels 4 and 5. Fig. 14 denotes the same sets f1; 2; 3g and f2; 3g as Fig. 13. The only difference is what set is considered as the universal set.
1 2 3 1 2
3
In the case where L = f apple, lemon, orange g and R = f lemon, orange g, 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.
a p ple lemon orange a p ple
lemon orange there is a bijection from X to an ordinal, thus if once we x such a bijection we may imagine a similar diagram for X. However, the general case of an in nite set X is not straightforward yet. 5
Beginners of sets and logic sometimes confuse exact de nition 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 nite set inclusion. We visualized the empty set by a closed glove.
As a by-product of our visualization media, we can visualize difference between \is an element of" with \is a subset of". Number i is an element of a given set exactly in the case where nger 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.
We are grateful to anonymous reviewers for helpful comments.