Question

1. State true or false.
(i) Rational numbers are closed under addition and multiplication.
(ii) Addition of rational numbers is closed if its sum is also a rational number.
(iii) Addition and multiplication of rational numbers is commutative but subtraction and division is not
commutative.
(iv) All rational numbers are integers.​

Answer 1

Answer:

1.FALSE

2.TRUE

3.TRUE

4.FALSE

Answer 2

The inferences for the above-given statements are as follows:

(i) False

• The quotient or fraction p/q of two numbers with a numerator of p and a non-zero denominator of q is called a rational number. Because q can equal 1, every integer is a rational number.
• The set of all rational numbers is known as the field of rationals, and it is usually abbreviated as Q.
• When an operation or a sequence of operations closes a set, the closure property is satisfied.
• The axiom of closure refers to the fact that a closure property is frequently represented as an axiom.
• Any two rational numbers, such as x and y, generate a rational number when they are added, subtracted, or multiplied.
• The closure characteristic does not cover division by zero.
• The outcome of addition, subtraction, and multiplication is a rational number. This indicates that in addition, subtraction, and multiplication, rational numbers are closed.

(ii) True

• The closure property asserts that a + b is a rational number for every two rational integers a and b. The ultimate result is a manageable figure. As a result, rational numbers are said to be closed when added.

(iii) True

• As a-b ≠ b-a and a÷b≠b÷a, hence, Subtraction and division are not commutative.

(iv) False

• Rational numbers are not integers as they are generally in the form of p/q, where, p and q are integers.