Mathematical Statement

Q.

A sentence is a statement if it can be said to be true or false, not both, without any ambiguity.

1. False

2. True

Q.

Write the negation of the following compound statement. 'There is a real number which is not a complex number'.

Q. 54 what are rational and irrational functions ?

Q.

Negation of "2+3 =5 and 8 is less than 10 " is

1. $2+3\ne 5and\text{8 is less than 10}$

2. $2+3\ne 5or\text{8 is less than 10}$

3. $2+3\ne 5and\text{8 is not less than 10}$

4. $2+3\ne 5or\text{8 is not less than 10}$

Q.

Identify the correct statement(s).

1. 'There exists' is the existential quantifier

2. 'For all' is the universal quantifier

3. 'For all' is the existential quantifier

4. 'There exists' is the universal quantifier

Q. Which of the following sentences are statements? Give reasons for your answer. (i) There are 35 days in a month. (ii) Mathematics is difficult. (iii) The sum of 5 and 7 is greater than 10. (iv) The square of a number is an even number. (v) The sides of a quadrilateral have equal length. (vi) Answer this question. (vii) The product of (–1) and 8 is 8. (viii) The sum of all interior angles of a triangle is 180°. (ix) Today is a windy day. (x) All real numbers are complex numbers.

Q.

Give three examples of sentences which are not statements. Give reasons for the answers.

(i) Who are you?

(ii) May God bless you!

(iii) How are you?

Q. There is a set $A$ possessing the following property
$"\forall x\in A\exists y\in A$ such that $y>x."$ This means

1. $y$ is always greater than $x.$
   
2. $x$ is the least element of $A.$
   
3. $A$ has no maxima.
   
4. Each element of the set is bigger than some other element.

Q.

Write the following in set-builder form:-i) Set of all +ve integers whose cube is odd ii) Set of all real numbers which cannot be written as the quotient of two integers

Q. Write the following statement in five different ways, conveying the same meaning. p: If triangle is equiangular, then it is an obtuse angled triangle.

Q. Check the validity of the following statements:
(i) p : 100 is a multiple of 4 and 5.
(ii) q : 125 is a multiple of 5 and 7.
(iii) r : 60 is a multiple of 3 or 5.