Re: [解題] 高一數學資優集合問題

看板tutor (家教)作者LeonYo (僕は美味しいです)時間10年前 (2015/04/03 07:50)推噓0(0推 0噓 4→)

留言4則, 1人參與討論串2/2 (看更多)

※ 引述《sunfin (遠方)》之銘言： : 1.年級：高一資優數學 : 2.科目：高中數學 : 3.章節：龍騰第一冊第一章集合問題 : 4.題目： ∞ ∞ ∞ ∞ ∞ Let {A_n} be a sequence of sets such that ∪ ∩ A_n = ∩ ∪ A_n. n=1 k=1 n=k k=1 n=k Which of the following sequenses has this property? (a) A_n = { x屬於R | -1/n < x <= 1+(-1)^n } (b) A_n = { x屬於R | -(1+n)/n < x < (1+n)/n } (c) A_n = { x屬於R | -(1+n)/n <= x <= (1+n)/n } [Sol]: (a) ∞ ∞ ∞ A_1 = (-1 , 0] ∩ A_n = [0,0], ∪ ∩ A_n =[0,0] A_2 = (-1/2, 2] n=k k=1 n=k A_3 = (-1/3, 0] A_4 = (-1/4, 2] ∞ ∞ ∞ A_5 = (-1/5, 0] ∪ A_n = (-1/k, 2], ∩ ∪ A_n =[0,2] A_6 = (-1/6, 2] n=k k=1 n=k This sequence has no such property. (b) ∞ ∞ ∞ B_1 = (-2 , 2 ) ∩ B_n = [-1,1], ∪ ∩ B_n =[-1,1] B_2 = (-3/2, 3/2) n=k k=1 n=k B_3 = (-4/3, 4/3) B_4 = (-5/4, 5/4) ∞ k+1 k+1 ∞ ∞ B_5 = (-6/5, 6/5) ∪ B_n = (---,---), ∩ ∪ B_n =[-1,1] B_6 = (-7/6, 7/6) n=k -k k k=1 n=k This sequence has this property. (c) ∞ ∞ ∞ C_1 = [-2 , 2 ] ∩ C_n = [-1,1], ∪ ∩ C_n =[-1,1] C_2 = [-3/2, 3/2] n=k k=1 n=k C_3 = [-4/3, 4/3] C_4 = [-5/4, 5/4] ∞ k+1 k+1 ∞ ∞ C_5 = [-6/5, 6/5] ∪ C_n = [---,---], ∩ ∪ C_n =[-1,1] C_6 = [-7/6, 7/6] n=k -k k k=1 n=k This sequence has this property. 這的確是 limsup 與 liminf 但只要對集合有基本的認識，老師都應該能看得懂並作出這題的答案要讓學生知道這題目在幹嘛很簡單就像我上面作的事情，一項一項帶進去給學生看，一項一項推導就好了「看不懂數列在幹什麼，就帶個幾項進去看看」這件事是一定要教給學生的 -- ※ 發信站: 批踢踢實業坊(ptt.cc), 來自: 114.42.229.80 ※ 文章網址: https://www.ptt.cc/bbs/tutor/M.1428018607.A.FD9.html

Re: [解題] 高一數學 資優集合問題

Re: [解題] 高一數學資優集合問題