a) Show that the relation R on Z x Z defined by (a , b) R (c, d) if and only
if a + d = b + c is an equivalence relation.
b) Show that a subset of an anti symmetric relation is also anti symmetric.
c) Suppose that R is a symmetric relation on a set A. Is R also symmetric?
Thank you!!
a) To prove that the relation R on Z x Z is an equivalence relation, we need to show that it satisfies three properties: reflexive, symmetric, and transitive.
1. Reflexive: For any element (a,b) in Z x Z, we need to show that (a,b) R (a,b). In this case, we have (a,b) R (a,b) if and only if a + b = b + a, which is true for all integers a and b. Therefore, R is reflexive.
2. Symmetric: For any elements (a,b) and (c,d) in Z x Z, if (a,b) R (c,d), then we need to show that (c,d) R (a,b). In this case, if a + d = b + c, then we have c + b = d + a, which implies (c,d) R (a,b). Therefore, R is symmetric.
3. Transitive: For any elements (a,b), (c,d), and (e,f) in Z x Z, if (a,b) R (c,d) and (c,d) R (e,f), then we need to show that (a,b) R (e,f). In this case, if a + d = b + c and c + f = d + e, then we have a + f = b + e, which implies (a,b) R (e,f). Therefore, R is transitive.
Since R satisfies all three properties (reflexive, symmetric, and transitive), it is an equivalence relation.
b) To show that a subset of an anti-symmetric relation is also anti-symmetric, we need to understand the definition of anti-symmetry first. An anti-symmetric relation R on a set A is said to be anti-symmetric if for any elements a and b in A, if (a,b) ∈ R and (b,a) ∈ R, then a = b.
Now, let's consider a subset S of an anti-symmetric relation R on set A. We want to show that for any elements a and b in S, if (a,b) ∈ S and (b,a) ∈ S, then a = b.
Since S is a subset of R, if (a,b) ∈ S, then (a,b) ∈ R. Similarly, if (b,a) ∈ S, then (b,a) ∈ R. Now, since R is anti-symmetric, if (a,b) ∈ R and (b,a) ∈ R, then a = b. Since S is a subset of R, the same (a,b) pairs will also be in S. Therefore, for any elements a and b in S, if (a,b) ∈ S and (b,a) ∈ S, then a = b.
Hence, we can conclude that a subset of an anti-symmetric relation is also anti-symmetric.
c) If R is a symmetric relation on a set A, it means that for any elements a and b in A, if (a,b) ∈ R, then (b,a) ∈ R. In other words, if (a,b) is related by R, then (b,a) is also related by R.
Therefore, if R is already a symmetric relation, there is nothing more to prove. Since the definition of symmetry is satisfied, we can conclude that R is symmetric.
In short, if a relation R is symmetric, then R is indeed symmetric.