Interval

1. Thm 1 :

1.1. 1 - | a | = a^2 , every a in R

1.2. 2- | a | < | b | a^2 < b^2 , every a , b in R

1.3. 3- | a.b | = | a | . | b | , every a , b in R

1.4. 4- if k > 0 then | a | < k -k < a < k , every a in R .

1.5. 5- | a+b | | a | + | b | , every a , b in R

1.6. 6- | a | - | b | | a - b | , every a , b in R

2. Absolute value

2.1. Note

2.1.1. The absolute value | a| of an element R is regarded as the distance from a , the origins more generally, the distance between element a and b is | a - b | .

3. Also can define

3.1. [ a , ) = { x in R : x a }

3.2. ( a , ) = { x in R : x > a }

3.3. ( - , a ] = { x in R : x a }

3.4. ( - , a ] = { x in R : x < a }

3.5. ( , ) = R .

4. Definition :

4.1. If a, b in R , a < b then we define

4.1.1. 1- An open interval ( a , b ) = { x in R : a < x < b}

4.1.2. 2- An closed interval [ a , b ] = { x in R : a x b }

4.1.3. 3- simi- open or simi closed interval ( a , b ] = { x in R : a < x b }

5. Define Min , Max

5.1. Let A R

5.1.1. 1- An element m in A , is said to be a minimum of A , if m a , every a in A . We write m=min A .

5.1.2. 2- An element M in m is said to be a maximum of A if M 0 , every a in A . We write m = Max A .