Question

設(shè)x、y∈R,求證:|x+y|=|x|+|y|成立的充要條件是xy≥0.

Answer 1

證明:充分性:若xy=0,那么,①x=0,y≠0;②x≠0,y=0;③x=0,y=0,于是|x+y|=|x|+|y|.如果xy＞0,即x＞0,y＞0或x＜0,y＜0;

當(dāng)x＞0,y＞0時(shí),|x+y|=x+y=|x|+|y|;

當(dāng)x＜0,y＜0時(shí),|x+y|=-(x+y)=-x+?(-y)=|x|+|y|.

總之,當(dāng)xy≥0時(shí),有|x+y|=|x|+|y|.

必要性:由|x+y|=|x|+|y|及x、y∈R,得

(x+y)²=(|x|+|y|)²,即x²+2xy+y²=x²+2|xy|+y².

|xy|=xy.∴xy≥0.

啟示:充要條件的證明關(guān)鍵是根據(jù)定義確定哪是已知條件,哪是結(jié)論,然后搞清楚充分性是證明哪一個(gè)命題,必要性是證明哪一個(gè)命題.