Question

explain the properties of mod and its rules for class 11

Answer 1

the absolute value of a number is called mod of that number. It is denoted by | |

condition for absolute value of a number x :
|x|= -x if x<0
|x|= x if x≥0

properties:
1. |xy|= |x| |y|
2. |x/y|= |x|/|y|
3. |x±y|≤ |x| + |y|
4. |x±y|≥ | |x| - |y| |
5. the inequality |x|≤ a means that -a≤x≤a
6. the inequality |x|≥a means that x≥a or x≤ -a

Answer 2

{\displaystyle |a|\geq 0}(2)Non-negativity{\displaystyle |a|=0\iff a=0}(3)Positive-definiteness{\displaystyle |ab|=|a|\,|b|}(4)Multiplicativity{\displaystyle |a+b|\leq |a|+|b|}(5)Triangle inequality#Example norms (equivalent to subadditivity)

The properties given by equations (2)-(4) are readily apparent from the definition. To see that equation (5) holds, choose {\displaystyle \varepsilon } from {\displaystyle \{-1,1\}} so that {\displaystyle \varepsilon (a+b)\geq 0}. Since {\displaystyle \varepsilon x\leq |x|} for real {\displaystyle x} regardless of the value of {\displaystyle \varepsilon }chosen, (5) follows from the calculation {\displaystyle |a+b|=\varepsilon (a+b)=\varepsilon a+\varepsilon b\leq |a|+|b|}. (For a generalization of this argument to complex numbers, see "Proof of the triangle inequality for complex numbers" below.)

Some additional useful properties are given below. These properties are either implied by or equivalent to the properties given by equations (2)-(5).

{\displaystyle {\big |}\,|a|\,{\big |}=|a|}(6)Idempotence (the absolute value of the absolute value is the absolute value){\displaystyle |-a|=|a|}(7)Evenness (reflection symmetry of the graph){\displaystyle |a-b|=0\iff a=b}(8)Identity of indiscernibles (equivalent to positive-definiteness){\displaystyle |a-b|\leq |a-c|+|c-b|}(9)Triangle inequality#Example norms (equivalent to subadditivity){\displaystyle \left|{\frac {a}{b}}\right|={\frac {|a|}{|b|}}\ } (if {\displaystyle b\neq 0})(10)Preservation of division (equivalent to multiplicativity){\displaystyle |a-b|\geq {\big |}\,|a|-|b|\,{\big |}}(11)Reverse triangle inequality (equivalent to subadditivity)