This question has 2 parts. Part 1: The drawing below shows a Hasse diagram for a partial order on the set: {А, В, С, D, E, F, G, H, 1, J} E H B Figure 1: A Hasse diagram shows 10 vertices and 8 edges. The vertices, represented by dots, are as follows: verter J is upward of verter H; verter H is upward of verter I; verter B is inclined upward to the left of verter A; verter C is upward of verter B; verter D is inclined upward to the right of verter C; verter E is inclined upward to the left of verter F; verter G is inclined upward to the right of verter E. The edges, represented by line segments between the vertices are as follows: 3 vertical edges connect the following vertices: B and C, H and I, and H and J; 5 inclined edges connect the following vertices: A and B, C and D, D and E, E and F, and E and G. Determine the properties of the Hasse diagram based on the following questions: (a) What are the minimal elements of the partial order? The minimal elements are J,I,A,F (b) What are the maximal elements of the partial order? (c) Which of the following pairs are comparable? (А, D), (J, F), (В, Е), (G, F), (D, в), (С, F), (Н, 1), (С, E)
Permutations and Combinations
If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.
Counting Theory
The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.
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