Recall from Definition 54.5 and Example 54.6 that the set of all partitions of a given
Chapter 0, Problem 12(choose chapter or problem)
Recall from Definition 54.5 and Example 54.6 that the set of all partitions of a given set, together with the refines relation, forms a poset. Please answer the following: a. Let P D f1; 2; 3; 4g; f5; 6; 7; 8; 9g and Q D f1; 3; 5; 7; 9g; f2; 4; 6; 8g . Calculate P ^ Q and P _ Q. b. Let P, Q, and R be partitions of an n-element set for which P D fX1; X2; : : : ; Xpg; Q D fY1; Y2; : : : ; Yqg; and R D P ^ Q D fZ1; Z2; : : : ; Zr g: Show that every Zk in R is of the form Xi \ Yj .
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