问题1.写以下语句的否定：
(i)班加罗尔是卡纳塔克邦的首都。
(ii)2005年7月4日下雨。
(iii)奢华是诚实的。
(iv)地球是圆的。
(v)太阳很冷。
解决方案：
(i) Bangalore is not the capital of Karnataka or It is false that “Bangalore is the capital of Karnataka.”
(ii) It did not rain on July 4, 2005 or It is false that it rained on July 4, 2005.
(iii) Ravish is dishonest or It is false that “Ravish is honest”.
(iv) The earth is not round or It is false that “The earth is round.”
(v) The sun is not cold or It is false that “The sun is cold.”
问题2：写以下语句的否定：
(i)所有的鸟都唱歌。
(ii)一些偶数是质数。
(iii)有一个复数，而不是实数。
(iv)我不会上学。
(v)矩形的两个对角线具有相同的长度。
(vi)所有警察都是小偷。
解决方案：
(i) All birds do not sing or It is false that “All birds sing.”
(ii) Not all even integers are prime or It is false that “even integers are prime.”
(iii) All complex number are real numbers or It is false that “complex numbers are not a real number.”
(iv) I will go to school.
(v) There is at least one rectangle whose both diagonals of unequal length.
(vi) No policemen are thief.
问题3.以下几对语句是否彼此相反?
(i)x不是有理数。
数字x不是无理数。
(ii)数字x不是有理数。
x是无理数。
解决方案：
(i) The number x is not a irrational number, means that the number x is a rational number.
Therefore, the second statement is negation of the first statement.
(ii) The number x is not a rational number means that the number x is an irrational number.
Therefore, the second statement is similar to the first statement, and therefore, they are not negation of each other.
问题4.编写以下语句的否定：
(i)p：对于每个正实数x，数字(x – 1)也是正数。
(ii)q：对于每个实数x，x> 1或x <1。
(iii)r：存在一个数字x，使得0
解决方案：
(i) We have,
p: For every positive real number x, the number (x – 1) is also positive.
The negation of the statement is,
~p: There exists at least one positive real number x, such that the number (x – 1) is not positive.
(ii) The negation of the statement:
~q: There exists at least one real number, s.t neither x>1 nor x<1.
(iii) The negation of the statement:
~r: For all real numbers x, such that either x ≤ 0 or x ≥ 1.
问题5.检查以下对语句是否彼此相反。解释你的回答。
(i)a + b = b + a对于每个实数a和b都是成立的。
(ii)存在实数a和b，其中a + b = b + a。
解决方案：
The negation of the (i) statement:
There exist real numbers are ‘a’ and ‘b’ for which a + b ≠ b + a.
So, (ii) is not negation of (i). Hence, these statements are not a negation of each other.