Conjunction and Disjunction Operations

Question

Let $ A_1 = \{1,2,3,6,7,14,21,42\}. $ Show that $A_1$ is closed under the operations $\lor$ and $\land$ defined by $a \land b = gcd(a,b)$ and $a \lor b = lcm(a,b).$

What are the identity elements in $A_1$ for $\lor$ and $\land$? Find the complement of each element in $A_1$.

I'm not too sure where to start on this question. Could I show the first part through the use of cayley tables and show that every element produced is a member of $A_1$?

Thanks in advance.

Answer 1

Closed under "$\wedge$" (GCD):

    |  1 |  2 |  3 |  6 |  7 | 14 | 21 | 42
----|----|----|----|----|----|----|----|----
  1 |  1 |  1 |  1 |  1 |  1 |  1 |  1 |  1
  2 |  1 |  2 |  1 |  2 |  1 |  2 |  1 |  2
  3 |  1 |  1 |  3 |  3 |  1 |  1 |  3 |  3
  6 |  1 |  2 |  3 |  6 |  1 |  2 |  3 |  6
  7 |  1 |  1 |  1 |  1 |  7 |  7 |  7 |  7
 14 |  1 |  2 |  1 |  2 |  7 | 14 |  7 | 14
 21 |  1 |  1 |  3 |  3 |  7 |  7 | 21 | 21
 42 |  1 |  2 |  3 |  6 |  7 | 14 | 21 | 42

Identity element is $42$.

---

Closed under "$\vee$" (LCM):

    |  1 |  2 |  3 |  6 |  7 | 14 | 21 | 42
----|----|----|----|----|----|----|----|----
  1 |  1 |  2 |  3 |  6 |  7 | 14 | 21 | 42
  2 |  2 |  2 |  6 |  6 | 14 | 14 | 42 | 42
  3 |  3 |  6 |  3 |  6 | 21 | 42 | 21 | 42
  6 |  6 |  6 |  6 |  6 | 42 | 42 | 42 | 42
  7 |  7 | 14 | 21 | 42 |  7 | 14 | 21 | 42
 14 | 14 | 14 | 42 | 42 | 14 | 14 | 42 | 42
 21 | 21 | 42 | 21 | 42 | 21 | 42 | 21 | 42
 42 | 42 | 42 | 42 | 42 | 42 | 42 | 42 | 42

Identity element is $1$.