GATE Overflow Test Series | Discrete Mathematics | Test 3 | Question: 13

Question

The following is the Hasse diagram of the poset $\left[\{1,2,3,4,6,7,12,14,21,28,42,84\},\:\mid\right],$ where $\mid $ is the "divides" relation.

If the number of complements of $1,3$ and $4$ are $\alpha,\beta$ and $\gamma$ respectively, then the value of $2\alpha + 3 \beta + 4\gamma $ is ________

Answer 1

Complement of an element $a$ is $a'$ if:

• $a \wedge a' = 0$ (lowest vertex in the Hasse diagram)
• $a \vee a' = 1$ (highest vertex in the Hasse diagram)

Complement of element $1$ is $84\implies \alpha  = 1$

Complement of element $3$ is $28\implies \beta = 1$

Complement of element $4$ is $21\implies \gamma = 1$

Now, $2\alpha + 3 \beta + 4\gamma = 2 + 3 + 4 = 9.$

Comments

yes it is

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All element has complement

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@mrinmoyh  yes these element dosnt have compement.