2.
Solve the following inequation, write the solution set and represent it on the number
line:
-x/3<x/3-4/3<1/6,x€R
Answers
Answer:
Explanation – According to above given question first, we have to find the poset for the divisibility.
Let the set is A.
A={(3
\prec 12), (3
\prec 24), (3
\prec 48), (3
\prec 72), (4
\prec 12), (4
\prec 24), (4
\prec 48), (4
\prec 72), (12
\prec 24), (12
\prec 48), (12
\prec 72), (24
\prec 48), (24
\prec 72)}
So, now the Hasse diagram will be:
example 1
In above diagram, 3 and 4 are at same level because they are not related to each other and they are smaller than other elements in the set. The next succeeding element for 3 and 4 is 12 i.e, 12 is divisible by both 3 and 4. Then 24 is divisible by 3, 4 and 12. Hence, it is placed above 12. 24 divides both 48 and 72 but 48 does not divide 72. Hence 48 and 72 are not joined.
We can see transitivity in our diagram as the level is increasing.
Example-2: Draw Hasse diagram for (D
_{12}, /)
Explanation – Here, D
_{12} means set of positive integers divisors of 12.
So, D
_{12}={1, 2, 3, 4, 6, 12}
poset A = {(1
\prec 2), (1
\prec 3), (1
\prec 4), (1
\prec 6), (1
\prec 12), (2
\prec 4), (2
\prec 6), (2
\prec 12), (3
\prec 6), (3
\prec 12), (4
\prec 12), (6
\prec 12)}
So, now the Hasse diagram will be-
example 2
In above diagram, 1 is the only element that divides all other elements and smallest. Hence, it is placed at the bottom. Then the elements in our set are 2 and 3 which do not divide each other hence they are placed at same level separately but divisible by 1 (both joined by 1). 4 is divisible by 1 and 2 while 6 is divisible by 1, 2 and 3 hence, 4 is joined by 2 and 6 is joined by 2 and 3. 12 is divisible by all the elements hence, joined by 4 and 6 not by all elements because we have already joined 4 and 6 with smaller elements accordingly.
Step-by-step explanation:
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