Suppose K = {{2,3,4,5,6,7},{2,5,7,8,12},{1,2,3,5,7,9,13,20}}. Find three upper bounds for K in P(N) and three lower bounds. Does K have a least upper bound? Does K has a greatest lower bound?
To find upper bounds for the set K in the power set of N (P(N)), we need to identify sets that contain all the elements of K. Similarly, to find lower bounds, we need to look for sets that are contained within K.
Let's begin with the upper bounds:
1. The universal set N (also denoted as U or P(N) itself) is an upper bound for K. This is because every element in K is a subset of N.
2. Another upper bound for K is the set {1, 2, 3, ..., 20}. This set contains all the elements from 1 to 20, which are the largest elements present in K.
3. Lastly, the set {2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 20} is an upper bound for K. It contains the individual elements that are the largest within their respective sets in K.
Moving on to the lower bounds:
1. The empty set {} is a lower bound for K because every set in K is a superset of the empty set.
2. The set {2, 5, 7} is another lower bound for K because it is a subset of every set within K.
3. Similarly, the set {2} is a lower bound for K as it is a subset of all the sets in K.
Now, let's address the questions about least upper bound and greatest lower bound:
K does not have a least upper bound in P(N). This is because there is no single set in P(N) that contains all the elements of K while being a subset of any other set that contains all the elements of K.
K also does not have a greatest lower bound in P(N). This is because there is no single set in P(N) that is a subset of all the sets in K while containing all the elements of any other set that is a subset of all the sets in K.
In summary, K does not have a least upper bound or a greatest lower bound in the power set of N (P(N)).