Reflexive, Transitive, Symmetric

A = {1,2,3}というSetがあったとき、以下を満たす集合Rを書いてみる。

• R is reflexive, symmetric, transitive
• R is reflexive, symmetric, but not transitive
• R is reflexive, transitive, but not symmetric
• R is reflexive, but not symmetric nor transitive
• R is not reflexive, nor symmetric, nor transitive
• R is not reflexive, not symmetric, but is transitive
• R is not reflexive, not transitive, but is symmetric
• R is not reflexive, but is symmetric and transitive

R is reflexive, symmetric, transitive

(1,1),(2,2),(3,3)

• (1,1),(2,2),(3,3) are reflexive because the R has the three pairs whose first coordinate and second coordinate are same, and so on.

R is reflexive, symmetric, but not transitive

(1,1),(1,2),(2,1),(2,2),(2,3),(3,2),(3,3)

• (1,2) and (2,3) are members of R, but (1,3) is not member of R.

R is reflexive, transitive, but not symmetric

(1,1),(1,2),(2,2),(3,3)

• (1,2) is member of R, but (2,1) is not member of R.

R is reflexive, but not symmetric nor transitive

(1,1),(1,2),(2,2),(2,3),(3,3)

• [not symmetric] (1,2) is member of R, but (2,1) is not member of R.
• [not transitive] (1,2) and (2,3) are members of R, but (1,3) is not member of R.

R is not reflexive, nor symmetric, nor transitive

(1,2),(2,3)

• [not relfexive] R doesn't have (2,2), (3,3).
• [not symmetric] (1,2) is member of R, but (2,1) is not member of R.
• [not transitive] (1,2) and (2,3) are members of R, but (1,3) is not member of R.

R is not reflexive, not symmetric, but is transitive

(1,2)

• [not relfexive] R doesn't have (1,1), (2,2), (3,3).
• [not symmetric] (1,2) is member of R, but (2,1) is not member of R.

not transitiveの理由
**for all a, b, c ∈ A, (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R, **

このstatementはimplicationというlogicで成り立っている。

implicationのTruth tableは以下。

A	B	A ⇒ B
T	T	T
T	F	F
F	T	T
F	F	T

(a, b) ∈ R and (b, c) ∈ R がFalseとなれば、全体としてのStatementはTrueになる。
よって、 R = {(1,2)}の場合、 (a, b) ∈ R and (b, c) ∈ R がFalseとなっており、transitiveはTrueとなる。

R is not reflexive, not transitive, but is symmetric

(1,2),(2,1)

• [not relfexive] R doesn't have (1,1), (2,2), (3,3).
• [not transitive] (1,2), (2,1) are members of R, but (1,1) is not member of R.

R is not reflexive, but is symmetric and transitive

(1,1),(1,2),(2,1),(2,2)

• [not relfexive] R doesn't have (3,3).