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AleksandrR [38]
2 years ago
14

Determine whether the relation o defined on ℤ is reflexive, symmetric, or transitive. The relation o on ℤ is defined as follows:

for all m, n ∈ ℤ, m o n ⇔ m – n is odd
Mathematics
1 answer:
Alexxx [7]2 years ago
5 0
  • Reflexive: no. A relation is reflexive is an element is in relation with itself. But in this case, it cannot happen, because x \circ x \iff x-x \text{ is odd} \iff 0 \text{is odd}, which clearly cannot happen
  • Symmetric: yes. A relation is symmetric if every time x and y are in relation, then also y and x are in relation. In this case, you have x and y are in relation if x-y is odd. But then, this guarantees that y and x are in relation, because y-x = -(x-y), and the opposite of an odd number is still odd
  • Transitive: no. A relation is transitive if every time x and y are in relation, and y and z are in relation, then x and z are in relation. So, suppose that x and y are in relation, which means x-y = 2k+1, for some integer k. We also know that y and z are in relation, which means that x-y = 2m+1, for some integer m. But then, you have

x-z = x-y+y-z = 2k+1 + 2m+1 = 2k+2m+2 = 2(k+m+1)

But since 2(k+m+1) is twice some integer, it is even, and thus x and z are not in relation. So, we've proven that although x \circ y and y \circ z, it can't be that x \circ z, and thus the relation is not transitive.

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