The following statement defines a relation R in the natural numbers N. state whether or not each relation is symmetric
1. x is less than or equal to y
2. x+y =10
To determine whether each relation is symmetric, we need to check if the reverse of each ordered pair is also an ordered pair in the relation.
1. Relation R: x is less than or equal to y
Let's analyze whether this relation is symmetric:
For any two values x and y in the set of natural numbers N, if x is less than or equal to y (x <= y), then we need to check if y is also less than or equal to x (y <= x).
Example:
Let x = 3 and y = 5
In this case, x <= y is true (3 <= 5).
However, y <= x is false (5 <= 3 is incorrect).
Since not all pairs (x, y) in relation R have the property that (y, x) is also in the relation, we can conclude that relation 1 is NOT symmetric.
2. Relation R: x + y = 10
Let's analyze whether this relation is symmetric:
For any two values x and y in the set of natural numbers N, if x + y = 10, then we need to check if y + x = 10.
Example:
Let x = 2 and y = 8
In this case, x + y = 10 (2 + 8 = 10).
Also, y + x = 10 (8 + 2 = 10).
Since for any pair (x, y) in relation R, the reverse pair (y, x) is also in the relation, we can conclude that relation 2 is symmetric.
In summary:
1. Relation R: x is less than or equal to y is NOT symmetric.
2. Relation R: x + y = 10 is symmetric.