Real numbers are closed with respect to addition and multiplication.
This means:
If you add or multiply real numbers the answer is also real.
Let's go through a couple of examples.
Closure Property of Real Number Addition
The problem 3 + 6 = 9 demonstrates the closure property of real number addition.
Observe that the addends and the sum are real numbers.
The closure property of real number addition states that when we add real numbers to other real numbers the result is also real.
In the example above, 3, 6, and 9 are real numbers
Closure Property of Real Number Multiplication
The problem 5 x 8 = 40 demonstrates the closure property of real number multiplication.
Observe that the factors and the product are real numbers.
The closure property of real number multiplication states that when we multiply real numbers with other real numbers the result is also real.
In the example above, 5, 8, and 40 are real numbers.
In general, Closure Property states that:
When you combine any two elements of the set the result is also in that set.
Real numbers are closed with respect to addition and multiplication. The examples above illustrate this.
but....what about subtraction and division? Are real numbers closed under subtraction and division too?
Well...subtraction of real numbers is closed but division of real numbers is not closed as we cannot divide by zero. There are situations when we don't get a closed system.
Subtraction of natural numbers is NOT closed.
Consider the natural numbers 7 and 8
7 - 8 = - 1
Negative 1 is NOT a natural number
So, closure property doesn't work here.
Therefore, the set of natural numbers is not closed under subtraction.
A. Real number
B. Irrational numbers
C. Rational numbers
D. Integers
Step 1: Here, only the set of irrational numbers does not satisfy closure property of addition.
Step 2: For example, consider the irrational numbers √12 and -√12
Step 3: √12 + (-√12√) = 0 is a rational number.
Step 4: So, the set of irrational numbers does not satisfy the Closure property under addition.
Step 1: 0, 11, and -11 are the elements of the given set {0, 11, - 11}.
Step 2: 0 x 11 = 0 [0 is an element of the set.]
Step 3: - 11 x 0 = 0 [0 is an element of the set.]
Step 4: - 11 x 11 = - 121, not an element of the given set.
Step 5: So, the given set does not satisfy the closure property with respect to multiplication.
Q1: Which of the following sets is NOT closed under addition?
Q2: Which of the following demonstrates the closure property of real numbers under multiplication?
Q: Why is division not closed for real numbers?
A: Because division by zero is undefined, and zero is a real number. Therefore, there exists at least one case where dividing two real numbers does not result in a real number.
Q: Is subtraction closed for natural numbers?
A: No, because subtracting a larger natural number from a smaller one results in a negative number, which is not a natural number (as demonstrated in the provided text).
Q: Are irrational numbers closed under addition?
A: No, as demonstrated by √12 + (-√12) = 0, where the sum of two irrational numbers is a rational number (0).