GO Classes Set Theory And Algebra Practice Set 1 | Question: 21

Question

Define the relation $\mathrm{O}$ on $\mathrm{Z}$ as follows:
$$
\forall m, n \in Z, m O n \longleftrightarrow \exists k \in Z \mid(m-n)=2 k+1
$$
Which one of the following statements about the relation $\mathrm{O}$ is true?

• A. The relation $\mathrm{O}$ is reflexive, not symmetric, and transitive.
• B. The relation $\mathrm{O}$ is reflexive, symmetric, and transitive.
• C. The relation $\mathrm{O}$ is not reflexive, not symmetric, and transitive.
• D. The relation $\mathrm{O}$ is not reflexive, symmetric, and not transitive.

Answer 1

Ans is D

for reflexivity

(m-m) =(2k-1) 

2k-1=0

k=½, which does not belong to Z

for symmetric

mOn is possible if m==n or m!=n will relate only if one is odd and another one even

if (m-n) = 2k-1 then (n-m)=2k-1

(n-m)=2k-1

-(m-n)=2k-1

so if in (m-n) we took k=x so for to hold (n-m) we just need to take k=-x

For transitivity

m-n=2k1-1, let m is odd and n is even

n-p=2k2-1  n is even and assume m is odd

---

m-p=2k3-1 so now m and p both are odd, so their subtraction will lead to even no . which can’t be written in 2k3-1 format where k3 belongs to z