Question
Find the least upper bound and greatest lower bound of $S=\{X,Y,Z\}$ if they exist, of the poset whose Hasse diagram is shown below:
The correct answer is
The least upper bound is Z and the greatest lower bound is E.
To find the least upper bound (LUB) and greatest lower bound (GLB) of the set \(S = \{X, Y, Z\}\) in the given poset from the Hasse diagram:
- Understanding the Poset:
- The diagram shows a partially ordered set (poset) with elements connected by lines indicating order.
- \(E\) is the minimal element and \(T, I\) are the topmost elements.
- Greatest Lower Bound (GLB):
- The GLB of a set is the greatest element that is less than or equal to every element of the set in the poset.
- For \(S = \{X, Y, Z\}\), the elements that are directly below any element in \(S\) should be considered.
- \(E\) is below both \(X\) and \(Y\), and is the lowest point where it reaches all elements in \(S\).
- Thus, the GLB of \(S\) is \(E\).
- Least Upper Bound (LUB):
- The LUB of a set is the least element that is greater than or equal to every element of the set in the poset.
- For \(S = \{X, Y, Z\}\), the elements that are directly above any element in \(S\) should be considered.
- \(Z\) is above both \(X\) and \(Y\) and encapsulates \(Z\) itself as well.
- Thus, the LUB of \(S\) is \(Z\).
Therefore, the least upper bound is \(Z\) and the greatest lower bound is \(E\).
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Important Questions from Set Theory and Venn Diagram (Notes)
Match List-I with List-II
List-1 List-II (A) If X and Y are two sets such that n(X)= 17, n(Y)=23, n(X $\cup$ Y)=38, then n(X $\cap$ Y) is (I) 20 (B)) If n(X) = 28,n(Y) = 32,n(X$\cap$Y) = 10, then n(X$\cup$Y) is (II) 10 (C) If n(X) = 10, then n(7X) is (III) 50 (D) If n(Y) = 20, then n($\frac{Y}{2}$) is (IV) 2 Choose the Correct answer from the options given below:
- From the given sets, which is an infinite set:
1. {x: x $\in$ N and (x-1)(x-2) = 0}
2. {x: x $\in$ N and x is prime number and less than 199}
3. {x: x $\in$ N and x$^5$ - 1 = 0}
4. {x: x $\in$ N and x is odd} Consider the following relation R={(4,5),(5,4), (7,6),(6,7)} on set I={4,5,6,7}. Which of the following properties relation R does not have?
A. Reflexive property
B. Symmetric property
C. Transitive property
D. Antisymmetric property
Choose the correct answer from the options given below:- In a class, 40 like Math, 35 like Science, 30 like English; 15 like both Math and Science, 12 like Math and English, 10 like Science and English, 5 like all three. How many like atleast one?
- Based on a survey of 200 students, 140 students like cold drinks, 120 students like milkshakes, and 80 students like both. How many students like at least one of the drinks ?
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