问题 填空题
若对任意x∈A,y∈B,(A⊆R,B⊆R)有唯一确定的f(x,y)与之对应,则称f(x,y)为关于x、y的二元函数.现定义满足下列性质的二元函数f(x,y)为关于实数x、y的广义“距离”;
(1)非负性:f(x,y)≥0,当且仅当x=y时取等号;
(2)对称性:f(x,y)=f(y,x);
(3)三角形不等式:f(x,y)≤f(x,z)+f(z,y)对任意的实数z均成立.
今给出三个二元函数,请选出所有能够成为关于x、y的广义“距离”的序号:
①f(x,y)=|x-y|;②f(x,y)=(x-y)2;③f(x,y)=
x-y

能够成为关于的x、y的广义“距离”的函数的序号是______.
答案

对于①,f(x,y)=|x-y|≥0满足(1),f(x,y)=|x-y|=f(y,x)=|y-x|满足(2);

f(x,y)=|x-y|=|(x-z)+(z-y)|≤|x-z|+|z-y|=f(x,z)+f(z,y)满足(3)

故①能够成为关于的x、y的广义“距离”的函数

对于②不满足(3)

对于③不满足(2)

故答案为①

单项选择题
阅读理解

My father was Chief engineer of merchant ship, which was sunk in Word War II. The book Night of the U-boats told the story.

Memories

In September, 1940, my mother, sister and I went to Swansea, where my father's ship was getting ready to sail. We brought him a family photograph to be kept with him at all times and keep him safe.

Then I remember my mother lying face down, sobbing. She had heard from a friend that the ship had been sunk by a torpedo (鱼雷).

I can remember the arrival of the telegram (电报), which in those days always brought had news. My grandmother opened it. It read, "Safe, Love Ted."

My most vivid memory is being woken and brought down to sit on my father's knee, his arm in a bandage.

He was judged unfit to return to sea and took a shore job in Glasgow for the rest of the war. For as long as I can remember, he had a weak heart. Mother said it was caused by the torpedoes. He said it was because of the cigarettes. Whichever, he died suddenly in his early 50s.

Ten years later I read Night of the U-boats and was able to complete the story.

Torpedo

One torpedo struck the ship. Father was in the engine room, where the third engineer was killed. He shut down the engines to slow the ship making it easier for it to be abandoned.

By the time he got on deck (甲板) he was alone. Every lifeboat was gone except one which had stuck fast. When he tried to cut it free, it swung against the ship, injuring his hand and arm. He had no choice but to jump — still with the photograph in his pocket.

Three days later, he and other survivors were safe in Glasgow. All 23 with him signed the back of the photograph.

A Toast

In my room is the book and the photograph. Often, glass in hand, I have wondered how I would have dealt with an explosion, a sinking ship, a jump into a vast ocean and a wait for rescue? Lest(以免)we forget, I have some more whisky and toast the heroes of the war.

小题1:We can infer that the mother and children went to Swansea ________.

A.to meet a friend

B.to see the father off

C.to take a family photo

D.to enjoy the sailing of the ship小题2:What did the author learn about the father from the telegram?

A.he was still alive.

B.His knee was broken.

C.His ship had been sunk.

D.He had arrived in Glasgow.小题3:The underlined word "it" in Paragraph 6 refers to the father's ________.

A.weak heart

B.taking a shore job

C.failure to return to sea

D.injury caused by a torpedo小题4:What can we know about the author's father after his ship was attacked?

A.He lost his arm

B.He repaired the engines.

C.He managed to take a lifeboat.

D.He was the last to leave the ship.小题5:What is the massage mainly about?

A.A group of forgotten heroes

B.A book describing a terrifying battle.

C.A ship engineer's wartime experience.

D.A merchant's memories of a sea rescue.