Let P D .X; / be a lattice. Let a; x1; x2; : : : ; xn 2 X and suppose that a xi for all 1 i n. Prove that a x1 ^ x2 ^ ^ xn.

1. RelatingDivisionandFractions a. Thecontextofaproblemwillchangethemeaningastowhetherweare examiningafractionorcompletingtheoperationofdividing. i. Example:2/5ofthecakerepresents2piecesofthewholecakeifitwere splitinto5equalpieces. ii. Contrast:2 ÷ 5istheamountofcakeeachpersonwillgetif2identical cakesaredividedequallyamong5people. b. Howdotheyrelate i. 1.Startwiththeideaofsplittingthecakesequallyamong5people 1. Todothis,wetakeeachcakeanddivideeachinto5equalpieces. Eachpersongetsonepiecefromeachcake ii. Now,takethatonepiecefromeachcakeandplaceitonanewcake. 1. Those3piecesmakeup2/5ofsinglecake. iii. Therefore,2/5=2÷5 iv. Ingeneralterms,A/B=A÷B 2. FractionswithNegativeNumbers a. Sowheneveryougetafractionwithanegativenumberinit,thereareafew waysthisproblemcango. i. Ifyouhavethenumeratorasthenegative,thewholefractionwillendup turningnegative. 1. Example:(-15/9)=-(15/19) ii. Ifyouhavethedenominatorasthenegative,thewholefractionwith ALSOendupturningnegative. 1. Example:(15/-9)=-(15/9) iii. BUT,ifyouhaveBOTHthenumeratorANDthe