Consider the relation R = (a,b),(a,c),(c,c),(b,b),(c,b),(b,c) on the set A = a,b. Is R reflexive? Symmetric? Transitive? and why or why not.
First off, you'd better have A={a,b,c} or the relation R is not over set A.
reflexive: (a,a)? No
Symmetric: (a,b) and (b,a)? No
Transitive: Yes, since
(a,b),(b,c) and (a,c)
(a,b),(b,b)
(a,c),(c,b) and (a,b)
(b,c),(c,b) and (b,b)
(b,c),(c,c)
(c,b),(b,c) and (c,c)
1235*845
To determine whether the relation R is reflexive, symmetric, or transitive, we need to understand what these terms mean in the context of relations.
Reflexive: A relation is reflexive if every element in the set has a relationship with itself. In other words, for every element 'x' in the set, (x, x) must be in the relation R.
Symmetric: A relation is symmetric if whenever (x, y) is in the relation R, then (y, x) is also in the relation R. In other words, if there is a relationship from 'x' to 'y', there must be a relationship from 'y' to 'x' as well.
Transitive: A relation is transitive if whenever (x, y) and (y, z) are in the relation R, then (x, z) is also in the relation R. In other words, if there is a relationship from 'x' to 'y' and a relationship from 'y' to 'z', then there must be a relationship from 'x' to 'z' as well.
Now, let's analyze the relation R given: R = {(a, b), (a, c), (c, c), (b, b), (c, b), (b, c)} on the set A = {a, b}.
1) Reflexivity: We check if every element in A has a relationship with itself.
- (a, a) is not in R.
- (b, b) is in R.
Since not every element has a relationship with itself, we can conclude that R is not reflexive.
2) Symmetry: We check if for every (x, y) in R, (y, x) is also in R.
- (a, b) is in R, but (b, a) is not in R.
- (a, c) is in R, but (c, a) is not in R.
- (c, c) is in R, and (c, c) is in R.
- (b, b) is in R, and (b, b) is in R.
- (c, b) is in R, and (b, c) is in R.
- (b, c) is in R, and (c, b) is in R.
Since there are elements in R where (x, y) is in R but (y, x) is not in R, we can conclude that R is not symmetric.
3) Transitivity: We check if for every (x, y) and (y, z) in R, (x, z) is also in R.
- (a, b) is in R, but there is no (b, ?) or (?, b) pair in R.
- (a, c) is in R, and (c, c) is in R, but there is no (a, ?) or (?, a) pair in R.
- (c, c) is in R, and (c, b) is in R, but there is no (c, ?) or (?, c) pair in R.
- (b, b) is in R, and (b, b) is in R, but there is no (b, ?) or (?, b) pair in R.
- (c, b) is in R, but there is no (b, ?) or (?, b) pair in R.
- (b, c) is in R, and (c, c) is in R, but there is no (b, ?) or (?, b) pair in R.
Since there are pairs (x, y) and (y, z) in R where (x, z) is not in R, we can conclude that R is not transitive.
In summary, the relation R is neither reflexive, symmetric, nor transitive.