Problem about Hasse diagrams

Question

Can someone help me to solve this problem.

Are these Hasse diagrams lattices?

Answer 1

(Following on the comments.) They're all latices. When you have a Hasse diagram, it's fairly easy to find greatest lower bounds and least upper bounds.

For instance, given $x,y$, if $x \le y$ then $x \vee y = y$ and $x \wedge y = x$. This is easy to spot because you can connect $x$ to $y$ by a path that moves in just one direction. In your first diagram, for example, you know that $a \le e$ because there's an upwards-directed path $a \to b \to e$.

Sometimes we don't have $x \le y$ or $y \le x$, e.g. $f$ and $g$ in your second diagram. To get from $f$ to $g$ you have to move up and down again, or down and up again, so they're not comparable. But that's still not a problem: just by looking at the diagram you can see that $f \vee g = h$ and $f \wedge g = b$.