Exercise 1.1 (Solutions)

Question 1(i)

Is the set $\{0\}$ has closure property w.r.t '+' or '$\times$'.

Solutions

Addition Table \[\begin{array}{|c|c|} \hline + & 0 \\ \hline 0 & 0 \\ \hline \end{array} \] As $0+0=0 \in \{0\}$.

This implies $\{0\}$ has closure property w.r.t. '+'.

Multiplication Table \[\begin{array}{|c|c|} \hline \times & 0 \\ \hline 0 & 0 \\ \hline \end{array} \]

As $0\times 0=0 \in \{0\}$

This implies $\{0\}$ has closure property w.r.t. '$\times$'.

Question 1(ii)

Is the set $\{1\}$ has closure property w.r.t. '+' or '$\times$'.

Solutions

Addition Table \[\begin{array}{|c|c|} \hline + & 1 \\ \hline 1 & 2 \\ \hline \end{array} \]

As $1+1=2 \notin \{1\}$.

This implies $\{1\}$ does not satisfy closure property w.r.t. '+'.

Multiplication Table \[\begin{array}{|c|c|} \hline \times & 1 \\ \hline 1 & 1 \\ \hline \end{array} \] As $1\times 1=1 \in \{1\}$. This implies $\{1\}$ has closure property w.r.t. '$\times$'

Question 1(iii)

Is the set $\{0,-1\}$ has closure property w.r.t. '+' or 'x'.

Solutions

Addition Table \[\begin{array}{|c|c|c|} \hline + & 0 & -1 \\ \hline 0 & 0 & -1 \\ \hline -1 & -1 & -2 \\ \hline \end{array} \] As $(-1)+(-1)=-2 \notin \{0,-1\}$.

$\Rightarrow \{0,-1\}$ does not satisfy closure property w.r.t. '+'.

Multiplication Table \[\begin{array}{|c|c|c|} \hline \times & 0 & -1 \\ \hline 0 & 0 & 0 \\ \hline -1 & 0 & 1 \\ \hline \end{array} \] As $(-1)\times (-1)= 1 \notin \{0,-1\}$.

$\Rightarrow \{0,-1\}$ does not have closure property w.r.t. '$\times$'.

Question 1(iv)

Is the set $\{1,-1\}$ has closure property w.r.t. '+' or '$\times$'.

Solutions

Addition Table \[\begin{array}{|c|c|c|} \hline + & 1 & -1 \\ \hline 1 & 2 & 0 \\ \hline -1 & 0 & -2 \\ \hline \end{array} \] As $1+1=2 \notin \{1,-1\}$.

$\Rightarrow \{1,-1\}$ does not closure property w.r.t. '+'.

Multiplication Table \[\begin{array}{|c|c|c|} \hline \times & 1 & -1 \\ \hline 1 & 1 & -1 \\ \hline -1 & -1 & 1 \\ \hline \end{array} \] As all the entries of the table belongs to $\{1,-1\}$.

$\Rightarrow \{1,-1\}$ has closure property w.r.t. '$\times$'.

Question 2 Name the properties used in the following equations (Letters, where used, represent real numbers).

Solutions

(i) $(4+9)=(9+4)$

Property: Commutative property w.r.t. '+'.

(ii) $(a+1)+ \frac{3}{4}= a+(1+\frac{3}{4})$

Property: Associative property w.r.t. '+'.

(iii) $(\sqrt{3}+\sqrt{5})+\sqrt{7}= \sqrt{3}(\sqrt{5}+\sqrt{7})$ used in the following question.

Property: Associative property w.r.t. '+'.

(iv) $(100+0) = 100$

Property: Additive identity w.r.t. '+'.

(v) $(1000 \times 1) = 1000$

Property: Multiplicative identity w.r.t. '$\times$'.

(vi) $4.1 +(-4.1) = 0$

Property: Additive inverse w.r.t. '+'.

(vii) $a - a = 0$

Property: Additive inverse w.r.t. '+'.

(viii) $\sqrt{2} \times \sqrt{5} = \sqrt{5} \times \sqrt{2}$

Property: Commutative property w.r.t. '$\times$'.

(ix) $a(b-c)= ab-ac$

Property: Left distributive property.

(x) $(x-y)z= xz-yz$

Property: Right distributive property.

(xi) $4 \times (5 \times 8)= (4 \times 5)\times 8$

Property: Associative property w.r.t. '$\times$'.

(xii) $a(b+c-d)= ab+ac-ad$

Property: Left distributive property.

Question 3 Name the properties used in the following inequalities:

Solution

(i) $-3<-2 \quad \Rightarrow 0<1$.

Property: Additive property.

(ii) $-5<-4 \quad \Rightarrow 20>16$.

Property: Multiplicative property.

(iii) $1>-1 \quad \Rightarrow -3>-5$

Property: Additive property.

(iv) $a<0 \quad \Rightarrow -a>0$

Property: Multiplicative property.

(v) $a>b \quad \Rightarrow \frac{1}{a}<\frac{1}{b}$.

Property: Multiplicative property.

(vi) $a>b \quad \Rightarrow -a<-b$.

Property: Multiplicative property.

Question 4(i)

Prove the following rules of addition: $$\frac{a}{c}+\frac{b}{c} = \frac{a+b}{c}$$.

Solution \begin{align} L.H.S &= \frac{a}{c}+\frac{b}{c}\\ &= a \times \frac{1}{c} + b \times \frac{1}{c}\\ &= (a+b) \times \frac{1}{c}\quad \text{(Right distributive property)}\\ &= \frac{a+b}{c}=R.H.S \end{align}

Question 4(ii)

Prove the following rules of addition: $$\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}$$.

Solution \begin{align} L.H.S &= \frac{a}{b}+\frac{c}{d}\\ &= \frac{a}{b} \times 1 + 1 \times \frac{c}{d}\\ &= \frac{a}{b} \times \left(d \times \frac{1}{d}\right)+\left(b \times \frac{1}{b}\right)\times \frac{c}{d}\\ &= \frac{a}{b} \times \frac{d}{d}+\frac{b}{b}\times \frac{c}{d}\\ &= \frac{ad}{bd} +\frac{bc}{bd}\\ &= ad \times \frac{1}{bd} +bc \times \frac{1}{bd}\\ &= (ad+bc) \times \frac{1}{bd} \\ &= \frac{ad+bc}{bd}=R.H.S \end{align}

Question 5

Prove that $$-\frac{7}{12}-\frac{5}{18}=\frac{-21-10}{36}$$.

Solution \begin{align} L.H.S &= -\frac{7}{12}-\frac{5}{18}\\ &= -\frac{7}{12} \times 1 - \frac{5}{18} \times 1\\ &= -\frac{7}{12} \times \left(3 \times \frac{1}{3}\right)- \frac{5}{18} \times \left(2 \times \frac{1}{2}\right) \\ &= -\frac{7}{12} \times \frac{3}{3}-\frac{5}{18}\times \frac{2}{2}\\ &= -\frac{21}{36} -\frac{10}{36}\\ &= -21 \times \frac{1}{36} -10 \times \frac{1}{36}\\ &= (-21-10) \times \frac{1}{36} \\ &= \frac{-21-10}{36}=R.H.S \end{align}

Question 6(i)

Simplify by justifying each step: $$\frac{4+16x}{4}$$

Solution

$\quad \dfrac{4+16x}{4}$

$=\dfrac{1}{4}\times (4+16x)\quad \because\dfrac{a}{b}=\dfrac{1}{b}\times a$

$=\dfrac{1}{4} \times (4\times 1+4 \times 4x)\quad \text{(multiplicative identity)}$

$=\dfrac{1}{4} \times 4 \times (1+ 4x)\quad \text{(distributive property)}$

$= 1 \times (1+4x)\quad \text{(multiplicative inverse)}$

$= 1+4x \quad \text{(multiplicative identity)}$

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