Question

8．已知x，y是实数，则“$\left\{\begin{array}{l}{x＞1}\\{y＞1}\end{array}\right.$”是$\left\{\begin{array}{l}{x+y＞2}\\{xy＞1}\end{array}\right.$的（　　）

• A． 充分不必要条件
• B． 必要不充分条件
• C． 充要条件
• D． 既不充分也不必要条件

Answer 1

分析 x，y是实数，则“$\left\{\begin{array}{l}{x＞1}\\{y＞1}\end{array}\right.$”⇒$\left\{\begin{array}{l}{x+y＞2}\\{xy＞1}\end{array}\right.$，反之不成立，例如：取x=4，y=$\frac{1}{2}$．即可判断出结论．

解答 解：∵x，y是实数，则“$\left\{\begin{array}{l}{x＞1}\\{y＞1}\end{array}\right.$”⇒$\left\{\begin{array}{l}{x+y＞2}\\{xy＞1}\end{array}\right.$，反之不成立，例如：取x=4，y=$\frac{1}{2}$．
∴则“$\left\{\begin{array}{l}{x＞1}\\{y＞1}\end{array}\right.$”是$\left\{\begin{array}{l}{x+y＞2}\\{xy＞1}\end{array}\right.$的充分不必要条件．
故选：A．

点评 本题考查了不等式的性质、简易逻辑的判定方法，考查了推理能力与计算能力，属于基础题．