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# 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).

## Discussion