充要条件习题课
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充要条件(习题课)
岳阳市四中：易亚红
知识回顾
1.若A=>B且B推不出A，则A是B的充分非必要条件
2.若A推不出B且B=>A，则A是B的必要非充分条件 3.若A=>B且B=>A，则A是B的充要条件 4.若A推不出B且B推不出A，则A既不是B的充分条件， 也不是B的必要条件.
练习一
1.已知p 是q 的必要而不充分条件，那么┐ p 是┐q的 ___ 2.若A是B的必要而不充分条件，C是B的充要条件，D 是C的充分而不必要条件，那么D是A的________ 3. 关于 x 的不等式：｜ x｜+｜x-1｜＞m 的解集为 R 的 充要条件是( ) (A)m＜0 (D)m≤1 答案： (1)充分不必要条件 (2)充分不必要条件 (3)C (B)m≤0 (C)m＜1
2.搞清①A是B的充分条件与A是B的充分非必 要条件之间的区别与联系；②A是B的必要条件与A 是B的必要非充分条件之间的区别与联系是非常重 要的,否则容易在这一点上出错误.
/ 合盛娱乐
ed by the Guardian of a Camp Fire circle in a small town in Pennsylvania to which she had belonged, the Akiyuhapi Camp Fire.” “The Are-you-happy Camp Fire! Sounds just like that!” put in Arline, rainbowed with mirthful memory. “Jessica told us that she had already been initiated a s a Wood Gatherer and showed her silver fagot ring. But we were a little flabbergasted, weren’t we, when she sprang her Indian name on us, by which she ha d chosen to be known among Camp Fire circles: Welatáwesit; it sounded musical as she pronounced it, but it seemed a mouthful! She partly explained it (d’you remember?) by saying that when she was choosing her symbolic name—as all Camp Fire Girls do —she wanted, for a special reason which she kept to herself, to take that of a flower, MorningGlory. And that Penobscot Indian word was the nearest she could get to it, the morning-glory not being originally a native plant .”“Yes, and it was at that very meeting, after we had welcomed Jessica with open arms as a Camp Fire Sister”—thus Sally again took up the fascinating thread of reminiscence—“that when each girl had told her symbolic name, Indian or otherwise, and how she came to choose it to express some special wish or aim, that we fell back upon digging for one for the new Camp Fire itself, the new circle or tribe. And then, don’t you remember”—Sesooā’s voice rose to a pi tch of excitement—“how Betty Ayres, little fair-haired Betty, who’s so enthusiastic and about as big as a minute—she’s just four feet, five inches and a half——”“My! but your minutes do stretch—like elastic,” put in Arline, with a rallying elbow poke.“Humph! Piffle! Betty jumped up suddenly as if she s aw a vision, with an idea swelling up so big in her that she seemed to grow two inches on the strength of it. ‘Girls!’ she cried, ‘I’m just tired of br owsing among Indian dictionaries, searching for a novel name for our new Camp Fire circle. Why don’t we call it, right away, the Morning -Glory Camp Fire? There’s a name that will reflect glory on us!’ said little Betty, half sobbing and half shining. ‘It suggests so much—so much that I can’t just put into words of —— ’”“‘Of the Morning of Life, the Glory of Girlhood—and vice versa—isn’t that what you mean, Betty dear?’ said our Guardian, helping her out!” This reminiscent contribution came from Arline. “And then Miss Dewey went on to say how she thought herself that it would be a glo rious name for us who are Daughters of the Sun, so to speak, having the Sun as our general symbol. So the Morning-Glory Camp Fire we are! And when we camp out this summer upon the Sugarloaf Peninsula where the sand-dunes are white as snow, we’re going to call our great, ramshackle wooden shanty, with one si de quite open to the airs of heaven, Camp Morning-Glory. So much glory that we shan’t know ourselves, eh? But all this”—slowly—“doesn’t bring us one little b it nearer to answering the question which I asked you at first, why our Glory-girl, Jessica, chose her symbolic name at the beginning. Since it put so much into our heads we’ve got a right to know all about it!” with another laughing stamp upon the playground grass. “I can’t bear mystery; if there’s a secret as big as my thumb, even if it’s about nothing or next to nothing, I want to know it.”“Oh, mystery—I love mystery! Bubbling mystery!” Sesooā rose on tiptoe under the Silver Twins, looking rather like a Baltimore oriole, that vivid flame-bird, for she, too, wore the latest thing in girlish smock frocks of a dainty peach-color very closely related to orange, shirred or smocked with black by her own clever little fingers that had fashioned the garment , too, the which had won her a green honor-bead to string upon the Camp Fire Girl’s necklace that she wore on ceremonial occasions.Those fingers had draped the little orange Tam O’Shanter, as well, which covered her crisp, dark hair, a masterpiece of head -gear more jaunty, less hood-like than that of the flower-like figure leaning against the auto’s side to which the wheeling firefly of her glance now turned.“Oh, bubbles! I’m going right over now to ask her why she chose her Morning-Glory name and symbol,” she went on, each word a tinted bubble of laughing curiosity painting itself upon the sunshine. “Absurd, but I am! If there’s any foolish little child-story woven in with the choice, this is the very time and place to hear it, here on the public playground, with all those children—such funny, foreign-looking tots most of them!—dancing ‘Pop Goes the Weasel!’ Pouf! I feel like dancing with them.”And the human oriole flitting forth from the friendly shade of the Twins fluttered her shirred plumage in a gleeful pas seul upon the playground grass, where the sun-glare transformed her into an orange flame, while her ears, attuned to all merry sounds, drank in the shrill music of five -and-thirty children’s voices (the number ought to have been even, but in that gleeful chorus there was one silent throat), six dancing sets, shouting with a st range babel of foreign accents, to the accompaniment of their stamping feet, the old nonsense-rhyme of the sixteenth century:“Half a pound of twopenny rice,Half a pound of treacle,Stir it up and make it nicePop goes the weasel!”musical scorehummed Sally, in flaming echo, and stood still.All the while, that versa tile quirk in her nature, corresponding to the flitting firefly in her eyes, which rendered her attention easily diverted when she wasn’t gravely in ea rnest, changed her all at once from an eager bubble of curiosity, that must burst if it did not penetrate a trifling secret, into an absorbed spectator. She hung upon the fringe of the playground dances, intent upon every rhythmic movement as the leading couple in each juvenile set (it happened to be a little earringed, lustrous-eyed Syrian girl footing it with a small Turk for a partner in that nearest) formed an arch with their uplifted arms for a gay little dan cer to pass beneath.“Oh-h! don’t they catch on well and dance prettily, these playground children?” murmured Sesooā softly to the quivering interest in her own heart. “I’m awfully glad that Jessica proposed our visiting this playground to
练习三
3 ．求关于 x 的方程 ax2+2x+1=0 至少有一个负的实根 的充要条件.
小结：本题解答时，一是容易漏掉讨论方程二次项 系数是否为零，二是只求必要条件忽略验证充分条 件.即以所求的必要条件代替充要条件.
课堂小结
1.在写某条件的充分或充要条件时，要特别 注意的是它们能否互相推出，切不可不加判断以 单向推出代替双向推出.
解：由题意知： 命题：若⌐p是⌐q的必要而不充分条件的等价命题即 逆否命题为：p是q的充分不必要条件. p:|1－(x-1)/3 |≤2 －2≤(x-1)/3 －1≤2 － 1≤(x-1)/3 ≤3 －2≤x≤10 q:x2－2x+1－m2≤0［x－(1－m)］［x－(1+m)］≤0 * ∵p是q的充分不必要条件， ∴不等式|1－ (x-1)/3 |≤2的解集是x2－2x+1－ m2≤0(m>0)解集的真子集. 又∵m>0 ∴不等式*的解集为1－m≤x≤1+m 3 m≥9， ∴ 1 m 2 m，∴ ∴实数m的取值范围是［9，+∞).
岳阳市四中：易亚红
知识回顾
1.若A=>B且B推不出A，则A是B的充分非必要条件
2.若A推不出B且B=>A，则A是B的必要非充分条件 3.若A=>B且B=>A，则A是B的充要条件 4.若A推不出B且B推不出A，则A既不是B的充分条件， 也不是B的必要条件.
练习一
1.已知p 是q 的必要而不充分条件，那么┐ p 是┐q的 ___ 2.若A是B的必要而不充分条件，C是B的充要条件，D 是C的充分而不必要条件，那么D是A的________ 3. 关于 x 的不等式：｜ x｜+｜x-1｜＞m 的解集为 R 的 充要条件是( ) (A)m＜0 (D)m≤1 答案： (1)充分不必要条件 (2)充分不必要条件 (3)C (B)m≤0 (C)m＜1
2.搞清①A是B的充分条件与A是B的充分非必 要条件之间的区别与联系；②A是B的必要条件与A 是B的必要非充分条件之间的区别与联系是非常重 要的,否则容易在这一点上出错误.
/ 合盛娱乐
ed by the Guardian of a Camp Fire circle in a small town in Pennsylvania to which she had belonged, the Akiyuhapi Camp Fire.” “The Are-you-happy Camp Fire! Sounds just like that!” put in Arline, rainbowed with mirthful memory. “Jessica told us that she had already been initiated a s a Wood Gatherer and showed her silver fagot ring. But we were a little flabbergasted, weren’t we, when she sprang her Indian name on us, by which she ha d chosen to be known among Camp Fire circles: Welatáwesit; it sounded musical as she pronounced it, but it seemed a mouthful! She partly explained it (d’you remember?) by saying that when she was choosing her symbolic name—as all Camp Fire Girls do —she wanted, for a special reason which she kept to herself, to take that of a flower, MorningGlory. And that Penobscot Indian word was the nearest she could get to it, the morning-glory not being originally a native plant .”“Yes, and it was at that very meeting, after we had welcomed Jessica with open arms as a Camp Fire Sister”—thus Sally again took up the fascinating thread of reminiscence—“that when each girl had told her symbolic name, Indian or otherwise, and how she came to choose it to express some special wish or aim, that we fell back upon digging for one for the new Camp Fire itself, the new circle or tribe. And then, don’t you remember”—Sesooā’s voice rose to a pi tch of excitement—“how Betty Ayres, little fair-haired Betty, who’s so enthusiastic and about as big as a minute—she’s just four feet, five inches and a half——”“My! but your minutes do stretch—like elastic,” put in Arline, with a rallying elbow poke.“Humph! Piffle! Betty jumped up suddenly as if she s aw a vision, with an idea swelling up so big in her that she seemed to grow two inches on the strength of it. ‘Girls!’ she cried, ‘I’m just tired of br owsing among Indian dictionaries, searching for a novel name for our new Camp Fire circle. Why don’t we call it, right away, the Morning -Glory Camp Fire? There’s a name that will reflect glory on us!’ said little Betty, half sobbing and half shining. ‘It suggests so much—so much that I can’t just put into words of —— ’”“‘Of the Morning of Life, the Glory of Girlhood—and vice versa—isn’t that what you mean, Betty dear?’ said our Guardian, helping her out!” This reminiscent contribution came from Arline. “And then Miss Dewey went on to say how she thought herself that it would be a glo rious name for us who are Daughters of the Sun, so to speak, having the Sun as our general symbol. So the Morning-Glory Camp Fire we are! And when we camp out this summer upon the Sugarloaf Peninsula where the sand-dunes are white as snow, we’re going to call our great, ramshackle wooden shanty, with one si de quite open to the airs of heaven, Camp Morning-Glory. So much glory that we shan’t know ourselves, eh? But all this”—slowly—“doesn’t bring us one little b it nearer to answering the question which I asked you at first, why our Glory-girl, Jessica, chose her symbolic name at the beginning. Since it put so much into our heads we’ve got a right to know all about it!” with another laughing stamp upon the playground grass. “I can’t bear mystery; if there’s a secret as big as my thumb, even if it’s about nothing or next to nothing, I want to know it.”“Oh, mystery—I love mystery! Bubbling mystery!” Sesooā rose on tiptoe under the Silver Twins, looking rather like a Baltimore oriole, that vivid flame-bird, for she, too, wore the latest thing in girlish smock frocks of a dainty peach-color very closely related to orange, shirred or smocked with black by her own clever little fingers that had fashioned the garment , too, the which had won her a green honor-bead to string upon the Camp Fire Girl’s necklace that she wore on ceremonial occasions.Those fingers had draped the little orange Tam O’Shanter, as well, which covered her crisp, dark hair, a masterpiece of head -gear more jaunty, less hood-like than that of the flower-like figure leaning against the auto’s side to which the wheeling firefly of her glance now turned.“Oh, bubbles! I’m going right over now to ask her why she chose her Morning-Glory name and symbol,” she went on, each word a tinted bubble of laughing curiosity painting itself upon the sunshine. “Absurd, but I am! If there’s any foolish little child-story woven in with the choice, this is the very time and place to hear it, here on the public playground, with all those children—such funny, foreign-looking tots most of them!—dancing ‘Pop Goes the Weasel!’ Pouf! I feel like dancing with them.”And the human oriole flitting forth from the friendly shade of the Twins fluttered her shirred plumage in a gleeful pas seul upon the playground grass, where the sun-glare transformed her into an orange flame, while her ears, attuned to all merry sounds, drank in the shrill music of five -and-thirty children’s voices (the number ought to have been even, but in that gleeful chorus there was one silent throat), six dancing sets, shouting with a st range babel of foreign accents, to the accompaniment of their stamping feet, the old nonsense-rhyme of the sixteenth century:“Half a pound of twopenny rice,Half a pound of treacle,Stir it up and make it nicePop goes the weasel!”musical scorehummed Sally, in flaming echo, and stood still.All the while, that versa tile quirk in her nature, corresponding to the flitting firefly in her eyes, which rendered her attention easily diverted when she wasn’t gravely in ea rnest, changed her all at once from an eager bubble of curiosity, that must burst if it did not penetrate a trifling secret, into an absorbed spectator. She hung upon the fringe of the playground dances, intent upon every rhythmic movement as the leading couple in each juvenile set (it happened to be a little earringed, lustrous-eyed Syrian girl footing it with a small Turk for a partner in that nearest) formed an arch with their uplifted arms for a gay little dan cer to pass beneath.“Oh-h! don’t they catch on well and dance prettily, these playground children?” murmured Sesooā softly to the quivering interest in her own heart. “I’m awfully glad that Jessica proposed our visiting this playground to
练习三
3 ．求关于 x 的方程 ax2+2x+1=0 至少有一个负的实根 的充要条件.
小结：本题解答时，一是容易漏掉讨论方程二次项 系数是否为零，二是只求必要条件忽略验证充分条 件.即以所求的必要条件代替充要条件.
课堂小结
1.在写某条件的充分或充要条件时，要特别 注意的是它们能否互相推出，切不可不加判断以 单向推出代替双向推出.
解：由题意知： 命题：若⌐p是⌐q的必要而不充分条件的等价命题即 逆否命题为：p是q的充分不必要条件. p:|1－(x-1)/3 |≤2 －2≤(x-1)/3 －1≤2 － 1≤(x-1)/3 ≤3 －2≤x≤10 q:x2－2x+1－m2≤0［x－(1－m)］［x－(1+m)］≤0 * ∵p是q的充分不必要条件， ∴不等式|1－ (x-1)/3 |≤2的解集是x2－2x+1－ m2≤0(m>0)解集的真子集. 又∵m>0 ∴不等式*的解集为1－m≤x≤1+m 3 m≥9， ∴ 1 m 2 m，∴ ∴实数m的取值范围是［9，+∞).