GATE Overflow Test Series | Discrete Mathematics | Test 3 | Question: 30

Question

The number of maximal and minimal elements for the poset $[\:\: \{2,4,6,9,12,18,27,36,48,60,72\},\: \mid\:\: ],$ where $\mid$ is the "divides" relation is

• A. $2,1$
• B. $4,1$
• C. $3,2$
• D. $4,2$

Answer 1

• Let $(A, R)$ be a poset. Then $'a'$ in $'A'$ is a minimal element if there does not exist another element $'b'$ in $A$ such that $bRa$.
•  Let $(A, R)$ be a poset. Then $'a'$ in $'A'$ is a maximal element if there does not exist another element $'b'$ in $A$ such that $aRb$.

$\textbf{Note:}$  There can be more than one minimal and maximal element in a poset (but exactly one each in a lattice).
 

The maximal elements are $27,60,48,72$ and minimal elements are $2,9.$

So, the correct answer is $(D)$.