Exercise 2.2 (Solutions)

Identify the property used in the following,

• (i) $a + b = b + a$ … …..
• (ii) $(ab)c = a(bc)$ … … …
• (iii) $7 \times 1 = 7$ … … …
• (iv) $x > y$ or $x = y$ or $x< y$ … … …
• (v) $ab = ba$ … … …
• (vi) $a + c = b + c \Rightarrow a = b$ … … …
• (vii) $5 + (-5) = 0$ … … …
• (viii) $7 \times \frac{1}{7} = 1$ … … …
• (ix) $a > b \Rightarrow ac > bc? (c >0)$ … … …

Solution

• (i) $a + b = b + a$ … …… ( Commutative w.r.t addition)
• (ii) $(ab)c = a(bc)$ … … … (Associative w.r.t multiplication)
• (iii) $7 \times 1 = 7$ … … … (Multiplicative w.r.t identity)
• (iv) $x > y$ or $x = y$ or $x< y$ … … … (Trichotomy)
• (v) $ab = ba$ … … … (Commutative w.r.t multiplicative)
• (vi) $a + c = b + c \Rightarrow a = b$ … … … (Cancellation property of addition)
• (vii) $5 + (-5) = 0$ … … … (Additive inverse)
• (viii) $7 \times \frac{1}{7} = 1$ … … … (Multiplicative inverse)
• (ix) $a > b \Rightarrow ac > bc?(c >0)$ … … … (Multiplicative property)

Fill in the following blanks by stating the properties of the real numbers used, $$ \begin{array}{cl} 3x + 3(y - x) &= 3x + 3y - 3x ... ... ... (i)\\ &= 3x - 3x + 3y ... ... ... (ii)\\ &= 0 + 3y ... ... ... (iii)\\ &= 3y ... ... ... (iv) \end{array} $$ Solution

• (i) Distributive property of multiplication over subtraction
• (ii) Commutative property
• (iii) Additive inverse
• (iv) Additive identity

Give the name of the property used in the following,

• (i) $\sqrt{24} + 0 = \sqrt{24}$ … … …
• (ii) $\frac{-2}{3} \left( 5 + \frac{7}{2}\right) = \left(\frac{-2}{3}\right){5} + \left(\frac{-2}{3}\right)\left(\frac{7}{2}\right)$ … …
• (iii) $\pi + \left(-\pi\right) = 0$ … … …
• (iv) $\sqrt{3} \times \sqrt{3}$ is real number … …. ….
• (v) $\left(\frac{-5}{8} \right )\left(\frac{-8}{5} \right ) = 1$ … … ….

Soluton

• (i) Additive identity.
• (ii) Distributive property of multiplication over addition .
• (iii) Additive inverse.
• (iv) Cllosure property of multiplication over addition.
• (v) Multiplicative inverse.