Answer to Question #348330 in Discrete Mathematics for kai

Question

Choose the best answer: ANTI SYMMETRIC, TRANSITIVE RELATION, SYMMETRIC RELATION, REFLEXIVE RELATION

1. {(3,5)(5,3) (2,4)(4,5)}

2. {(3,1)(1,3)}

3. {(3,1)(2,3)(5,6)(6,5)}

4. {(1,3)(5,3)(7,5)}

5. {(8,9)(9,7)(8,7)}

6. {(4,2)(2,4)}

7. {(1,5)(2,5)(3,5)}

8. {(5,5)(5,6)(6,5)(6,6)(6,7)(7,6) (7,7)}

9. {(6,5)(5,4)(6,4)}

10. {(7,6)(6,5)(7,5)}

11. {(3,1)(1,3)}

12. {(4,3)(3,5)(4,5)}

13. {(6,6)(6,5)(5,5)(5,4)(4,4)}

14. {(6,5)(7,6)(4,5)(5,4)}

15. {(3,3)(4,4)(4,5)(5,4)(5,5)}

Answer 1

1. {(3,5)(5,3) (2,4)(4,5)}

We have $(2,4)∈R$ but $(4,2)\not∈R,$ thus $R$ is not symmetric.

We have $(3,5)∈R, (5,3)∈R$ but $3\not=5,$ thus $R$ is not antisymmetric.

Since $(3,5)∈R$ and $(5,3)∈R,$ but $(3,3)∉R$ the relation $R$ is not transitive.

Since $(2,2),(3,3),(4,4), (5,5)∉R,$ the relation $R$ is irreflexive, hence, it is not reflexive.

2. {(3,1)(1,3)}

We have $(3,1)∈R, (1,3)∈R,$ thus $R$ is symmetric.

We have $(3,1)∈R, (1,3)∈R$ but $3\not=1,$ thus $R$ is not antisymmetric.

Since $(3,1)∈R$ and $(1,3)∈R,$ but $(3,3)∉R$ the relation $R$ is not transitive.

Since $(1,1),(3,3)∉R,$ the relation $R$ is irreflexive, hence, it is not reflexive.

3. {(3,1)(2,3)(5,6)(6,5)}

We have $(3,1)∈R$ but $(1,3)\not∈R,$ thus $R$ is not symmetric.

We have $(5,6)∈R, (6,5)∈R$ but $5\not=6,$ thus $R$ is not antisymmetric.

Since $(2,3)∈R$ and $(3,1)∈R,$ but $(2,1)∉R$ the relation $R$ is not transitive.

Since $(1,1),(2,2),(3,3),(5,5), (6,6)∉R,$ the relation $R$ is irreflexive, hence, it is not reflexive.

4. {(1,3)(5,3)(7,5)}

We have $(1,3)∈R$ but $(3,1)\not∈R,$ thus $R$ is not symmetric.

There is no pair of elements $a$ and $b$ with $a ≠ b$ such that both$(a, b)$ and $(b, a)$ belong to the relation. Thus $R$ is antisymmetric.

Since $(7,5)∈R$ and $(5,3)∈R,$ but $(7,3)∉R$ the relation $R$ is not transitive.

Since $(1,1),(3,3),(5,5), (7,7)∉R,$ the relation $R$ is irreflexive, hence, it is not reflexive.

5. {(8,9)(9,7)(8,7)}

We have $(8,9)∈R$ but $(9,8)\not∈R,$ thus $R$ is not symmetric.

There is no pair of elements $a$ and $b$ with $a ≠ b$ such that both$(a, b)$ and $(b, a)$ belong to the relation. Thus $R$ is antisymmetric.

Since $(8,9)∈R$ and $(9,7)∈R,$ and $(8,7)\in R$ the relation $R$ is transitive.

Since $(7,7),(8,8),(9,9)∉R,$ the relation $R$ is irreflexive, hence, it is not reflexive.

6. {(4,2)(2,4)}

We have $(4,2)∈R, (2,4)∈R,$ thus $R$ is symmetric.

We have $(2,4)∈R, (4,2)∈R$ but $2\not=4,$ thus $R$ is not antisymmetric.

Since $(4,2)∈R$ and $(2,4)∈R,$ but $(4,4)∉R$ the relation $R$ is not transitive.

Since $(2,2),(4,4)∉R,$ the relation $R$ is irreflexive, hence, it is not reflexive.

7. {(1,5)(2,5)(3,5)}

We have $(1,5)∈R$ but $(5,1)\not∈R,$ thus $R$ is not symmetric.

There is no pair of elements $a$ and $b$ with $a ≠ b$ such that both$(a, b)$ and $(b, a)$ belong to the relation. Thus $R$ is antisymmetric.

The relation $R$ is transitive.

Since $(1,1),(2,2),(3,3), (5,5)∉R,$ the relation $R$ is irreflexive, hence, it is not reflexive.

8. {(5,5)(5,6)(6,5)(6,6)(6,7)(7,6) (7,7)}

The relation $R$ is symmetric.

We have $(5,6)∈R, (6,5)∈R$ but $5\not=6,$ thus $R$ is not antisymmetric.

The relation $R$ is transitive.

The relation $R$ is reflexive.

9. {(6,5)(5,4)(6,4)}

We have $(5,4)∈R$ but $(4,5)\not∈R,$ thus $R$ is not symmetric.

There is no pair of elements $a$ and $b$ with $a ≠ b$ such that both$(a, b)$ and $(b, a)$ belong to the relation. Thus $R$ is antisymmetric.

The relation $R$ is transitive.

Since $(4,4),(5,5),(6,6), ∉R,$ the relation $R$ is irreflexive, hence, it is not reflexive.

10. {(7,6)(6,5)(7,5)}

We have $(7,6)∈R$ but $(6,7)\not∈R,$ thus $R$ is not symmetric.

There is no pair of elements $a$ and $b$ with $a ≠ b$ such that both$(a, b)$ and $(b, a)$ belong to the relation. Thus $R$ is antisymmetric.

The relation $R$ is transitive.

Since $(5,5),(6,6),(7,7), ∉R,$ the relation $R$ is irreflexive, hence, it is not reflexive.

11. {(3,1)(1,3)}

We have $(3,1)∈R, (1,3)∈R,$ thus $R$ is symmetric.

We have $(1,3)∈R, (3,1)∈R$ but $3\not=1,$ thus $R$ is not antisymmetric.

Since $(3,1)∈R$ and $(1,3)∈R,$ but $(3,3)∉R$ the relation $R$ is not transitive.

Since $(1,1),(3,3)∉R,$ the relation $R$ is irreflexive, hence, it is not reflexive.

12. {(4,3)(3,5)(4,5)}

We have $(4,3)∈R$ but $(3,4)\not∈R,$ thus $R$ is not symmetric.

There is no pair of elements $a$ and $b$ with $a ≠ b$ such that both$(a, b)$ and $(b, a)$ belong to the relation. Thus $R$ is antisymmetric.

The relation $R$ is transitive.

Since $(3,3),(4,4), (5,5)∉R,$ the relation $R$ is irreflexive, hence, it is not reflexive.

13. {(6,6)(6,5)(5,5)(5,4)(4,4)}

We have $(6,5)∈R,$ but $(5,6)\not∈R,$ thus $R$ is not symmetric.

The relation $R$ is antisymmetric.

Since $(6,5)∈R, (5,4)∈R$ but $(6,4)\not∈R,$ the relation $R$ is not transitive.

Since $(4,4), (5,5), (6,6),\in R,$ the relation $R$ is reflexive.

14. {(6,5)(7,6)(4,5)(5,4)}

We have $(6,5)∈R,$ but $(5,6)\not∈R,$ thus $R$ is not symmetric.

We have $(4,5)∈R, (5,4)∈R$ but $4\not=5,$ thus $R$ is not antisymmetric.

Since $(6,5)∈R, (5,4)∈R$ but $(6,4)\not∈R,$ the relation $R$ is not transitive.

Since $(4,4), (5,5),(6,6), (7,7)∉R,$ the relation $R$ is irreflexive, hence, it is not reflexive.

15. {(3,3)(4,4)(4,5)(5,4)(5,5)}

The relation $R$ is symmetric.

We have $(4,5)∈R, (5,4)∈R$ but $4\not=5,$ thus $R$ is not antisymmetric.

The relation $R$ is transitive.

Since $(3,3),(4,4), (5,5)\in R,$ the relation $R$ is reflexive.