问题1：否定以下陈述：
(i)p：对于每个正实数x，数字x – 1也是正数。
解决方案：
~p: There exists atleast a positive real number x, such that x – 1 is not positive.
(ii)q：所有的猫都抓挠。
解决方案：
~q: There exists cats that do not scratch.
(iii)r：对于每个实数x，x> 1或x <1。
解决方案：
~r: There exists a real number x, such that neither x > 1 nor x < 1.
(iv)s：存在一个数字x，使得0
解决方案：
~s: There does not exist a number x, such that 0 < x < 1.
问题2：陈述以下各陈述的相反和对立：
(i)p：一个正整数只有在除以1和本身以外没有除数时才是质数。
解决方案：
Statement p can be understood as follows.
If a positive integer is prime, then it has no divisors other than 1 and itself.
The converse of the statement is as follows.
If a positive integer has no divisor other than 1 and itself, then it is prime.
The contrapositive of the statement is as follows.
If positive integer has divisor other than 1 and itself, then it is not prime.
(ii)问：每当晴天时，我都会去海滩。
解决方案：
The given statement can be understood as follows.
If it is a sunny day, then I go to a beach.
The converse of the statement is as follows.
If I go to a beach, then it is a sunny day.
The contrapositive of the statement is as follows.
If I do not go to a beach, then it is not a sunny day.
(iii)r：如果外面很热，那么您会感到口渴。
解决方案：
The converse of statement r is as follows.
If you feel thirsty, then it is hot outside.
The contrapositive of statement r is as follows.
If you do not feel thirsty, then it is not hot outside.
问题3：以“如果p，则q”的形式编写每个语句。
(i)p：必须具有密码才能登录到服务器。
解决方案：
p: If you have a password, then you can log on to the server.
(ii)问：下雨时交通堵塞。
解决方案：
q: If it rains, then there is a traffic jam.
(iii)r：只有支付订阅费，您才能访问该网站。
解决方案：
r: If you pay the subscription fee, then you can access the website.
问题4：当且仅当q时，将以下每个语句重写为“ p”形式。
(i)p：如果您看电视，那么您的思想是自由的，如果您的思想是免费的，那么您就看电视。
解决方案：
p: You watch television if and only if your mind is free.
(ii)问：要获得A级成绩，有必要定期做所有作业。
解决方案：
q: You get an A grade if and only if you do all the homework regularly.
(iii)r：如果四边形是等角的，那么它是一个矩形，如果四边形是矩形的，那么它是等角的。
解决方案：
r: A quadrilateral is equiangular if and only if it is a rectangle.
问题5：以下是两个陈述
p：25是5的倍数。
q：25是8的倍数。
编写将这两个语句与“ And”和“ Or”连接起来的复合语句。在这两种情况下，都要检查复合语句的有效性。
解决方案：
The compound statement with ‘And’ is ‘25 is a multiple of 5 and 8′.
This statement is not valid, because 25 is not a multiple of 8.
The compound statement with ‘Or’ is ‘25 is a multiple of 5 or 8’.
This statement is valid, because although 25 is not a multiple of 8, it is a multiple of 5.
问题6：通过针对其的方法检查以下给出的陈述的有效性。
(i)p：无理数和有理数之和是无理的(通过矛盾方法)。
解决方案：
The given statement is,
p: The sum of an irrational number and a rational number is irrational.
Let us assume that the given statement, p, is false. That is, we assume that the sum of an irrational number and
a rational number is rational.
Therefore, √a + b/c = d/e is irrational where a, b, c, d and e are integers.
d/e – b/c is a rational number and √a is an irrational number.
This is a contradiction. So, our assumption is wrong.
Therefore, the sum of an irrational number and a rational number is rational.
Hence, the given statement is valid.
(ii)q：如果n是n> 3的实数，则n 2 > 9(通过矛盾方法)。
解决方案：
The given statement is,
q: If n is a real number with n > 3, then n2 > 9.
Let’s assume that n is a real number with n > 3, but n2 > 9 is false,i.e., n2 < 9.
Then, n > 3 where n is a real number.
Squaring both the sides,
n2 > (3)2
⇒ n2 > 9, which is a contradiction to our assumption that is n2 < 9.
Therefore, the given statement is valid.
问题7：用五种不同的方式写以下语句，传达相同的含义。
p：如果一个三角形是等角的，则它是一个钝角三角形。
解决方案：
The given statement can be written in the following five different ways:
(i) A triangle is equiangular implies that it is obtuse-angled.
(ii) A triangle is equiangular only if it is an obtuse-angled.
(iii) For a triangle to be equiangular, it is necessary that the triangle is obtuse-angled.
(iv) For a triangle to be obtuse-angled, it is sufficient that the triangle is equiangular.
(v) If a triangle is not obtuse-angled, then the triangle can not be equiangular.