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Integers Test - 22

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Integers Test - 22
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  • Question 1
    1 / -0
    Set of integers is not closed under ……..
    Solution
    Integers are closed under addition, subtraction, and multiplication operations. But the division of two integers need not be an integer.

    Example:-
    $$\dfrac12=0.5$$
    Here, $$1$$ and $$2$$ are integers but $$0.5$$ is not.
  • Question 2
    1 / -0
    Which of the following statements is incorrect?
    Solution
    When an integer is subtracted from another integer, the difference obtained is also an integer. Thus, integers are closed under subtraction.
    Similarly multiplication of two integers is an integer itself and addition of two integers is an integer.
  • Question 3
    1 / -0
    The product $$15\times 17 $$ is equal to
    Solution
    Whole numbers are commutative under multiplication. Therefore, $$15\times 17 = 17\times 15$$.
  • Question 4
    1 / -0
    The addition $$13 + 253$$ is equal to
    Solution
    Whole numbers are commutative under addition. therefore, $$13 + 253 = 253 + 13$$.
  • Question 5
    1 / -0
    Which of the following expression proves that integers are not closed under division?
    Solution
    $$1\div 2 = \dfrac {1}{2}$$
    Integers are whole numbers in addition to the negatives of natural numbers.
    $$\therefore \dfrac {1}{2}$$ is not an integer where as $$1$$ and $$2$$ are both integers. Hence, proved that integers are not closed under division.
  • Question 6
    1 / -0
    Positive integers are not closed under .............
    Solution
    Positive integers are numbers from $$1, 2, 3, ....$$ 

    So, in other words, they are natural numbers. And we know that natural numbers are closed under addition and multiplication only. So, positive integers are not closed under subtraction.

    For example: Consider $$5$$ and $$8$$

    Adding $$5$$ and $$8,$$ 

    $$5+8=13$$ and $$8+5=13$$

    So, positive integers are closed under addition.

    Now, let us multiply $$5$$ and $$8,$$ 

    $$5\times 8=40$$ and $$8\times 5=40$$

    So, positive integers are closed under multiplication.

    Now, let us find the difference between $$5$$ and $$8,$$ 

    $$5- 8=-3$$ and $$8- 5=3$$

    So, positive integers are not closed under subtraction.
  • Question 7
    1 / -0
    Positive integers are closed under ............
    Solution
    Positive integers are numbers from $$1, 2, 3, 4, .......$$ 
    So, they are natural numbers in other words. And we know, natural numbers are closed under addition and multiplication only.
  • Question 8
    1 / -0
    Positive and negative integers together are closed under ............
    Solution
    The set of positive and negative integers does not include $$0$$. Now, addition of two integers may result in $$0.$$ 
    Eg : $$-1 + 1 = 0$$ 
    And subtraction of two integers may also result in $$0.$$
    Eg : $$-1 - (-1) = -1 + 1 = 0.$$ 
    And we also know that integers are not closed under division. Also, when two non-zero numbers are multiplied, the product is a non-zero integer.
  • Question 9
    1 / -0
    Integers are commutative under ..............
    Solution
    Commutative property for addition:
    Integers are commutative under addition when any two integers are added irrespective of their order, the sum remains the same.
    a+b =b+a
    The sum of two integer numbers is always the same. This means that integer numbers follow the commutative property.

    Let’s see the following examples:
    15 + 20 =35; 20 +15=35
    -10 + (-5) = -15; -5 + (-10) = -15

    The above examples prove that the addition of integers is commutative. 

    The commutative property for Subtraction:
    Is the case true with subtractions? Are subtractions also commutative? The following examples will let us know this:
    5-(-3) = +8
    -3-5 = -8
    This brings us to the conclusion that subtractions of integers are not commutative. Therefore, a-b ≠ b-a

    Commutative Property of Division
    This property does not apply to divisions between integers. This means that a÷b ≠b÷a

    Option A is correct
  • Question 10
    1 / -0
    Integers are commutative under ........
    Solution
    For any two integers $$a$$ and $$b, a + b = b + a$$ and $$a\times b = b\times a$$.
    Thus, integers are commutative under addition and multiplication.
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