Question

3. Let S = {1,2,...,7,8) be ordered as in figure below. Consider the subset A = {3,6,7} of S. a. Find the set of upper bounds of A b. Find the set of lower bounds of A C. Does sup(A) exist? d. Does inffA) exist? 4. Repeat problem 3 for the subset B = {1,2,4,7) of S. 5. Let S be the ordered set in figure below. Suppose A = {1,2,3,4,5) is order-isomorphic to S and the following is a similarity mapping from Sonto A : f = {(2,1),(6,4),(c,5),(d, 2), (,3)} Draw the Hasse diagram of A.

please 3&4&5

Answer 1

(3) We have given:

$S = \left \{ 1, 2, ..., 7, 8 \right \}, A = \left \{ 3, 6, 7 \right \}$

(a)

Since every number in A is less than or equal to 7, so 7 is upper bound of A.

Therefore, every number in S greater than or equal to 7 is upper bound for A.

Set of upper bounds of A = {7, 8}

(b)

Similarly every number in A is greater than or equal to 3, so 3 is lower bound of A.

Therefore, every number in S leass than or equal to 3 is lower bound of A.

Set of lower bounds of A = {1, 2, 3}

(c) least upper bound of A is called as supremum of A.

Here least upper bound of A is 7.

Therefore, sup(A) = 7.

(d) greatest lower bound is called infimum of A.

Here greatest lower bound of A is 3.

Therefore, inf(A) = 3.