The relation R on {1,2,3,...} where aRb means a/b for reflexive, symmetric, antisymmetric or transitive in the binary family.
Also, the relation R on the set of all pepole where aRB means that a is at least as tall as b. Where is the releation R on N where aRb means that a has the same number of digits as b.
For the relation R on {1,2,3,...} where aRb means a/b:
Reflexive property: The relation R is reflexive if every element a is related to itself. In this case, every number a can be divided by itself (a/a = 1), so the relation R is reflexive.
Symmetric property: The relation R is symmetric if whenever a is related to b, then b is related to a. In this case, if a/b, then b/a (since division is commutative). Therefore, the relation R is symmetric.
Antisymmetric property: The relation R is antisymmetric if whenever both aRb and bRa, then a = b. In this case, if a/b and b/a, then a = b (since division is commutative). Therefore, the relation R is antisymmetric.
Transitive property: The relation R is transitive if whenever aRb and bRc, then aRc. In this case, if a/b and b/c, then a/c (since division is transitive). Therefore, the relation R is transitive.
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For the relation R on the set of all people where aRB means that a is at least as tall as b:
Reflexive property: The relation R is reflexive if every element a is related to itself. In this case, every person a is at least as tall as themselves. Therefore, the relation R is reflexive.
Symmetric property: The relation R is symmetric if whenever a is related to b, then b is related to a. In this case, if a is at least as tall as b, then b is at least as tall as a. Therefore, the relation R is symmetric.
Antisymmetric property: The relation R is antisymmetric if whenever both aRB and bRA, then a = b. In this case, if a is at least as tall as b and b is at least as tall as a, then a and b must have the same height. Therefore, the relation R is antisymmetric.
Transitive property: The relation R is transitive if whenever aRB and bRC, then aRC. In this case, if a is at least as tall as b and b is at least as tall as c, then a is at least as tall as c. Therefore, the relation R is transitive.
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For the relation R on N where aRb means that a has the same number of digits as b:
Reflexive property: The relation R is reflexive if every element a is related to itself. In this case, every number a has the same number of digits as itself. Therefore, the relation R is reflexive.
Symmetric property: The relation R is symmetric if whenever a is related to b, then b is related to a. In this case, if a has the same number of digits as b, then b has the same number of digits as a. Therefore, the relation R is symmetric.
Antisymmetric property: The relation R is antisymmetric if whenever both aRb and bRa, then a = b. In this case, if a has the same number of digits as b and b has the same number of digits as a, then a and b must be equal. Therefore, the relation R is antisymmetric.
Transitive property: The relation R is transitive if whenever aRb and bRc, then aRc. In this case, if a has the same number of digits as b and b has the same number of digits as c, then a has the same number of digits as c. Therefore, the relation R is transitive.
To determine whether a relation is reflexive, symmetric, antisymmetric, or transitive, we need to understand each of these properties.
1. Reflexive: A relation R is reflexive if every element is related to itself. In other words, for every element 'a' in the set, (a,a) must be in the relation. To check reflexivity, you need to verify if (a,a) is true for all 'a' in the given set.
For the relation R on {1,2,3,...}, where aRb means a/b, we can see that for any element 'a', a/a = 1, which is true. Hence, the relation R is reflexive.
2. Symmetric: A relation R is symmetric if for every (a,b) in the relation, (b,a) is also in the relation. To check symmetry, you need to verify if for every (a,b) that satisfies the relation, (b,a) also satisfies the relation.
For the relation R on {1,2,3,...}, where aRb means a/b, we can see that if a/b is true, then b/a is also true because division is commutative. Hence, the relation R is symmetric.
3. Antisymmetric: A relation R is antisymmetric if for any distinct elements 'a' and 'b' in the set, if (a,b) is in the relation, then (b,a) cannot be in the relation. To check antisymmetry, you need to verify if (a,b) being true implies that (b,a) is false whenever a ≠ b.
For the relation R on {1,2,3,...}, where aRb means a/b, we can see that if a ≠ b, then a/b ≠ b/a. Hence, the relation R is antisymmetric.
4. Transitive: A relation R is transitive if for every (a,b) and (b,c) in the relation, (a,c) is also in the relation. To check transitivity, you need to verify if for every (a,b) and (b,c) that satisfy the relation, (a,c) also satisfies the relation.
For the relation R on {1,2,3,...}, where aRb means a/b, we can see that if a/b and b/c are true, then a/c is also true because division is transitive. Hence, the relation R is transitive.
Regarding the relation R on the set of all people, where aRb means a is at least as tall as b, it is not explicitly mentioned whether the relation is reflexive, symmetric, antisymmetric, or transitive. In general, height comparison is not reflexive (as people aren't taller than themselves), not antisymmetric (as two different people could have the same height), and not transitive (as height comparisons can have cycles, e.g., if person A is taller than B, B is taller than C, but C is taller than A). However, it is symmetric because if person A is at least as tall as person B, then person B is at least as tall as person A.
Regarding the relation R on N (the set of natural numbers) where aRb means that a has the same number of digits as b, it is reflexive, symmetric, and transitive but not antisymmetric. To check each property, you need to verify the conditions mentioned earlier for reflexivity, symmetry, antisymmetry, and transitivity. In this case, it can be easily observed that the relation satisfies reflexivity, symmetry, and transitivity, as numbers have the same number of digits as themselves, digit equality is symmetric, and digits do not form a partial ordering.