Sets, Relations & Functions
Properties of Relations (Reflexive, Symmetric, Transitive)
nta_pyq_2023_jan
Grade 11
Question:
The relation R = \{(a, b) : \gcd(a, b) = 1, 2a \neq b, a, b \in \mathbb{Z}\} is:
transitive but not reflexive
symmetric but not transitive
reflexive but not symmetric
neither symmetric nor transitive
Step-by-Step Solution
Key Concept: Check each property: for reflexive, gcd(a,a)=|a| which equals 1 only if a=$\pm$1; for symmetric, check if (a,b)$\in$R implies (b,a)$\in$R; for transitive, find a counterexample.
Reflexive: (a,a)$\Rightarrow$gcd(a,a)=1 which is not true for every a$\in$Z. Not reflexive. Symmetric: Take a=2,b=1$\Rightarrow$gcd(2,1)=1, 2a=4$\neq$b; now a=1,b=2$\Rightarrow$gcd(1,2)=1, 2a=2=b, so (1,2)$\notin$R. Hence R is not symmetric. Transitive: a=14,b=19,c=21: gcd(a,b)=1, gcd(b,c)=1 but gcd(a,c)=7. Not transitive. Answer: (4)
Correct Answer: 4