Consider the divisibility relation on the set S = {-5,-3,-2,2,3,5}
To be more precise, this is the relation:
R = {(x, y) ∈ S^2| x divides y}.
Is the relation Reflexive? Symmetric? Anti-symmetric? Transitive?
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The relation is reflexive because all inputs of x is related to itself.
The relation is not symmetric because x/y =/= y/x
Would it be anti-symmetric?
For transitive, I understand that if xRy, yRz, then xRz, but what would the z values be?
Also, what are the x and y values for this? Or is it for any values of x and y? I don't get where the x and y inputs come from. Would the x and y values be one of the values from {-5,-3,-2,2,3,5}?
To determine if the relation R is anti-symmetric, we need to check if whenever (x, y) and (y, x) are both in R, it implies that x = y. In other words, if x divides y and y divides x, then x must equal y.
In this case, if we consider the pairs (-2, 2) and (2, -2), we see that -2 divides 2 and 2 divides -2, but -2 is not equal to 2. Therefore, the relation R is not anti-symmetric.
For the transitive property, we need to verify if whenever (x, y) and (y, z) are both in R, it implies that (x, z) is also in R. In other words, if x divides y and y divides z, then x must divide z.
Taking into account the pairs (-5, -5), (-5, 5), (-3, -3), (-3, 3), (-2, -2), (-2, 2), (2, 2), and (3, 3), we can see that for any combination of these pairs where y is not 0, the relation holds true. This implies that the relation R is transitive.
The x and y values in the relation R are chosen from the set S = {-5, -3, -2, 2, 3, 5}. So, any pair (x, y) in R must have x and y as elements of this set.
In summary:
- The relation R is reflexive.
- The relation R is not symmetric.
- The relation R is not anti-symmetric.
- The relation R is transitive.
To determine if the relation is anti-symmetric, we need to check if whenever (x, y) and (y, x) are in R, then x = y. In this case, (x, y) means x divides y.
For example, let's take x = -5 and y = 5. In this case, (-5, 5) is in R because -5 divides 5. However, (5, -5) is not in R because 5 does not divide -5. So, we have (x, y) in R, but not (y, x). This implies that the relation is not anti-symmetric.
To determine if the relation is transitive, we need to check if whenever (x, y) and (y, z) are in R, then (x, z) is also in R. In other words, if x divides y and y divides z, then x divides z.
For example, let's take x = 2, y = -2, and z = 4. In this case, (2, -2) is in R because 2 divides -2, and (-2, 4) is in R because -2 divides 4. However, (2, 4) is not in R because 2 does not divide 4. So, in this case, we have (x, y) and (y, z) in R, but not (x, z). This implies that the relation is not transitive.
Regarding x and y values, they can be any values from the set S = {-5, -3, -2, 2, 3, 5}. This means that x and y can take any of these values, and the relation R is defined based on whether x divides y or not.