Question #348330

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)}


1
Expert's answer
2022-06-06T15:00:08-0400

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

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

We have (3,5)R,(5,3)R(3,5)∈R, (5,3)∈R but 35,3\not=5, thus RR is not antisymmetric.

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

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


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

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

We have (3,1)R,(1,3)R(3,1)∈R, (1,3)∈R but 31,3\not=1, thus RR is not antisymmetric.

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

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


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

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

We have (5,6)R,(6,5)R(5,6)∈R, (6,5)∈R but 56,5\not=6, thus RR is not antisymmetric.

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

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


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

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

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

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

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


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

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

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

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

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


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

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

We have (2,4)R,(4,2)R(2,4)∈R, (4,2)∈R but 24,2\not=4, thus RR is not antisymmetric.

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

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


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

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

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

The relation RR is transitive.

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


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

The relation RR is symmetric.

We have (5,6)R,(6,5)R(5,6)∈R, (6,5)∈R but 56,5\not=6, thus RR is not antisymmetric.

The relation RR is transitive.

The relation RR is reflexive.



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

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

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

The relation RR is transitive.

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


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

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

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

The relation RR is transitive.

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


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

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

We have (1,3)R,(3,1)R(1,3)∈R, (3,1)∈R but 31,3\not=1, thus RR is not antisymmetric.

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

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


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

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

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

The relation RR is transitive.

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


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

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

The relation RR is antisymmetric.

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

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


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

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

We have (4,5)R,(5,4)R(4,5)∈R, (5,4)∈R but 45,4\not=5, thus RR is not antisymmetric.

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

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


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

The relation RR is symmetric.

We have (4,5)R,(5,4)R(4,5)∈R, (5,4)∈R but 45,4\not=5, thus RR is not antisymmetric.

The relation RR is transitive.

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



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