Let \$N\$ denote the set of all natural numbers and \$R\$ be the relation on \$N\times N\$ defined by \$(a, b) R (c, d)\Longleftrightarrow ad (b+c)=bc(a+d)\$. Check weather \$R\$ is an equivalence relation.

Let R be defined on \$N\times N\$ as \$(a, b) R (c, d)\Longleftrightarrow ad (b+c)=bc(a+d)\$. ....(1)Reflexivity:We can write \$ab(b+a)= ba(a+b)\$ for all \$a, b\in N\$Since, sum and product of natural numbers obeys commutative propertyHence, by def (1), we can write \$(a,b) R (a,b)\$ for all \$(a,b) \in N\times N\$Hence, \$R\$ is reflexive.Symmmetry :Let \$(a,b) R (c,d)\$ \$\Rightarrow ad(b+c)=bc(a+d)\$\$\Rightarrow da(c+b)=cb(d+a)\$ (Since, sum and product of natural numbers obeys commutative property)or \$cb(d+a)=da(c+b)\$\$\Rightarrow (c,d) R(a,b)\$Hence, \$R\$ is symmetric Transitivity :Let \$(a,b) ,(c,d) , (e,f) \in N\times N\$Let \$(a,b)R (c,d)\$ and \$(c,d)R(e,f)\$\$ad(b+c)=bc(a+d)\$ and \$cf(d+e)=de(c+f)\$\$\Rightarrow \dfrac{ab}{a-b} =\dfrac{cd}{c-d}\$ and \$\dfrac{cd}{c-d}=\dfrac{ef}{e-f}\$\$\Rightarrow \dfrac{ab}{a-b}=\dfrac{ef}{e-f}\$\$\Rightarrow af(b+e) =be(a+f)\$\$\Rightarrow (a,b)R(e,f)\$Hence ,\$R\$ is transitive\$\therefore R\$ is Equivalence Relation.

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>>Class 6
>>Maths
>>Whole Numbers
>> $Properties of Whole Numbers$

Properties of Whole Numbers

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Properties of Whole Number Operations

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Q: The property represented by \$a \times (b + c) = a \times b + a \times c\$ is

Distributive property say that multiplication an division can be distributed over parenthesis.a x (b+c) = a x b + a x cSo correct answer will be option C

Q: Which of the following statements is incorrect?

The difference of two whole numbers is not necessarily a whole number. The statement whole numbers are closed under subtraction is an incorrect statement.

Q: Find the value of the following.\$3845\times 5\times 782+769\times 25\times 218\$.

\$3845\times 782+769\times 25\times 218\$\$=3845\times 5\times 782+769\times 5\times 5\times 218\$\$=3845\times 5\times 782+3845\times 5\times 218\$\$=3845\times 5\times (782+218)\$\$=19225\times 1000=19225000\$.

Q: A student multiplied \$7236\$ by \$65\$ instead of multiplying by \$56\$. By how much was his answer greater than the correct answer?

(Hint: Do you need to do both the multiplications?)Difference between \$65\$ and \$56=9\$Difference in the answer\$=7236\times 9\$\$7236\$\$\times 9\$_________\$65124\$_________Therefore, His answer was greater than the correct answer by \$65,124\$.

Q: Find the product using suitable properties.\$854\times 102\$.

\$854\times 102=854\times (100+2)\$\$=854\times 100+854\times 2\$ \$(Distributive\$ \$property)\$\$=85400+1708\$\$=87108\$.

Q: Find the value of the following.\$81265\times 169-81265\times 69\$.

\$81265\times 169-81265\times 69\$\$=81265\times (169-69)\$\$=81265\times 100\$\$=8126500\$.

Q: Verify whether commutative property is satisfied for addition, subtraction, multiplication and division of the following pairs of rational numbers.(i) \$4\$ and \$\dfrac {2}{5}\$(ii) \$\dfrac {-3}{7}\$ and \$\dfrac {-2}{7}\$

(i)Addition\$4+\dfrac{2}{5}=\dfrac{20+2}{5}=\dfrac{22}{5}\$\$\dfrac{2}{5}+4=\dfrac{2+20}{5}=\dfrac{22}{5}\$Commutative property is satisfied.Subtraction\$4-\dfrac{2}{5}=\dfrac{20-2}{5}=\dfrac{18}{5}\$\$\dfrac{2}{5}-4=\dfrac{2-20}{5}=\dfrac{-18}{5}\$Commutative property is not satisfied.Multiplication\$\dfrac{2}{5}\times 4=\dfrac{8}{5}\$\$4\times \dfrac{2}{5}=\dfrac{8}{5}\$Commutative property is satisfied.Division\$4\div \dfrac{2}{5}=4\times \dfrac{5}{2}=\dfrac{20}{2}\$\$\dfrac{2}{5}\div 4=\dfrac{2}{5}\times \dfrac{1}{4}=\dfrac{2}{20}\$Commutative property is not satisfied.(ii)Addition\$\dfrac{-3}{7}+\left ( \dfrac{-2}{7} \right )=\dfrac{-3-2}{7}=\dfrac{-5}{7}\$\$\dfrac{-2}{7}+\left ( \dfrac{-3}{7} \right )=\dfrac{-2-3}{7}=\dfrac{-5}{7}\$Commutative property is satisfied.Subtraction\$\dfrac{-3}{7}-\left ( \dfrac{-2}{7} \right )=\dfrac{-3+2}{7}=\dfrac{-1}{7}\$\$\dfrac{-2}{7}-\left ( \dfrac{-3}{7} \right )=\dfrac{-2+3}{7}=\dfrac{1}{7}\$Commutative property is not satisfied.Multiplication\$\dfrac{-2}{7}\times \dfrac{-3}{7}=\dfrac{6}{7}\$\$\dfrac{-3}{7}\times \dfrac{-2}{7}=\dfrac{6}{7}\$Commutative property is satisfied.Division\$\dfrac{-3}{7}\div \dfrac{-2}{7}=\dfrac{-3}{7}\times \dfrac{7}{-2}=\dfrac{3}{2}\$\$\dfrac{-2}{7}\div \dfrac{-3}{7}=\dfrac{-2}{7}\times \dfrac{7}{-3}=\dfrac{2}{3}\$Commutative property is not satisfied.

Q: Find the product \$8739 \times 102\$ using distributive property.

\$8739\times 102\$\$=8739\times(100+2)\$\$=8739\times100+8739\times2\$\$=873900+17478\$\$=891,378\$

Q: Find the value of the following.\$54279\times 92+8\times 54279\$.

\$54279\times 92+8\times 54279\$\$=54279\times 92+54279\times 8\$\$=54279\times (92+8)\$\$=54279\times 100\$\$=5427900\$.

Properties of Natural and Whole Numbers

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Find the value of the following.
$54279 \times 92 + 8 \times 54279$ .

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Revise with Concepts

Properties of Whole Number Operations

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Quick Summary With Stories

Closure Property of Whole Numbers.

Commutative Properties of Whole Numbers.

Associative Property of Whole Numbers

One as a Multiplicative Identity.

Properties of Whole Number Operations

Additive and Multiplicative Identity of Whole Numbers

Let N denote the set of all natural numbers and R be the relation on N×N defined by (a,b)R(c,d)⟺ad(b+c)=bc(a+d). Check weather R is an equivalence relation.

The property represented by a×(b+c)=a×b+a×c is

Which of the following statements is incorrect?

Find the value of the following. 3845×5×782+769×25×218.

A student multiplied 7236 by 65 instead of multiplying by 56. By how much was his answer greater than the correct answer?

Find the product using suitable properties. 854×102.

Find the value of the following. 81265×169−81265×69.

Verify whether commutative property is satisfied for addition, subtraction, multiplication and division of the following pairs of rational numbers. (i) 4 and 52​ (ii) 7−3​ and 7−2​

Find the product 8739×102 using distributive property.

Find the value of the following. 54279×92+8×54279.

Let $N$ denote the set of all natural numbers and $R$ be the relation on $N \times N$ defined by $(a, b) R (c, d) ⟺ a d (b + c) = b c (a + d)$ . Check weather $R$ is an equivalence relation.

The property represented by $a \times (b + c) = a \times b + a \times c$ is

Find the value of the following.
$3845 \times 5 \times 782 + 769 \times 25 \times 218$ .

A student multiplied $7236$ by $65$ instead of multiplying by $56$ . By how much was his answer greater than the correct answer?

Find the product using suitable properties.
$854 \times 102$ .

Find the value of the following.
$81265 \times 169 - 81265 \times 69$ .

Verify whether commutative property is satisfied for addition, subtraction, multiplication and division of the following pairs of rational numbers.
(i) $4$ and $\frac{2}{5}$
(ii) $\frac{- 3}{7}$ and $\frac{- 2}{7}$

Find the product $8739 \times 102$ using distributive property.

Find the value of the following.
$54279 \times 92 + 8 \times 54279$ .