第 11 类 RD Sharma 解决方案 – 第 31 章导数 – 练习 31.1
问题 1. 找出下列句子中哪些是陈述，哪些不是。证明你的答案。
(i) 听我说，拉维！
(ii) 每个集合都是有限集合。
(iii) 两个非空集合总是有一个非空交集。
(iv) 猫的阴部是黑色的。
(v) 所有的圆都是圆的吗？
(vi) 所有三角形都有三个边。
(vii) 每个菱形都是正方形。
(viii) x 2 + 5|x| + 6 = 0 没有实根。
(ix) 这句话是陈述句。
(x) 地球是圆的吗？
（十一）去吧！
(xii) 实数 x 小于 2。
(xiii) 一个月有 35 天。
(xiv) 数学难。
(xv) 所有实数都是复数。
(xvi) (-1) 和 8 的乘积是 8。
解决方案：
Note: Any declarative sentence which can have either an affirmative(true/ yes) or negative (false/no), but not both answers can be considered as a statement or a proposition.
(i) Exclamatory sentences are not statements. Therefore, the line, “Listen to me, Ravi!” is not a statement.
(ii) There exists infinite sets also, and therefore, this sentence is false. Hence, it is a statement.
(iii) This sentence will always hold a false value, because there are non-empty sets whose intersection is empty. Therefore, it is a statement.
(iv) Some of the cats are not black, hence, this sentence may have a yes or no answer depending on the cat. Hence, it is not a statement.
(v) Interrogative sentences are not statements. Therefore, the question, “Are all circles round?” is not a statement.
(vi) Every triangle has three sides, therefore, this is a declarative statement with a true value.
(vii) A rhombus with not all equal sides are not squares. Therefore, this sentence is false and hence a statement,
(viii) Solving the equation, we have,
For x > 0
x2 + 5 |x| + 6 = 0
⇒ x2 + 5x + 6 = 0
We obtain,
⇒ x = – 3 or x = – 2
But, since x > 0, the equation has no roots.
For x < 0
x2 + 5 |x| + 6 = 0
⇒ x2 – 5x + 6 = 0
There are no possible real roots, which makes the sentence to be always true. Hence, it is a statement.
(ix)The sentence “This sentence is a statement: cannot be assigned a truth value of either true or false. Because either assignment contradicts the sense of the sentence.
(x) Interrogative sentences are not statements. Therefore, the question, “Is the earth round?” is not a statement.
(xi) Exclamatory sentences are not statements. Therefore, the line, “Go!” is not a statement.
(xii) The sentence may hold either true or false value depending on the value of x. Therefore, it is not a statement, because its value can’t be known without knowing the value of x.
(xiii) There are either 30 or 31 days in a month, therefore, this statement is false.
(xiv) Some people may find mathematics difficult, and some may like it. Therefore, the true or false value is dependent on the person and hence, it is not a statement.
(xv) This sentence is always true. Hence, it is a statement.
(xvi) This sentence is always false, because the product of -1 and 8 is -8. Hence, it is a statement.
问题 2. 给出三个不是陈述句的例子。给出答案的理由。
解决方案：
A statement or a proposition is an assertive (or a declarative) sentence which is either true or false but not both.
Example 1: “Aha! What a bird ! ” – Exclamatory sentences are not statements. Therefore, this line is not a statement.
Example 2: “Are you enrolled on GFG?” – Interrogative sentences are not statements. Therefore, this question, is not a statement.
Example 3: “Girls are more sweet than boys” – This depends on the each person’s outlook, hence it may vary. Therefore, it is not a statement.