Lecture 1 - Fundamental Concepts PDF

LECTURE 1 FUNDAMENTAL CONCEPTS BASIC LAW OF NATURAL NUMBERS Let a, b and c be any number. 1. Law of closure for addition a + b=z Example: 2 + 5 = 7 2. Commutative law for addition a+b=b+a Example: 5 + 7 = 7 + 5...

LECTURE 1 FUNDAMENTAL CONCEPTS BASIC LAW OF NATURAL NUMBERS Let a, b and c be any number. 1. Law of closure for addition a + b=z Example: 2 + 5 = 7 2. Commutative law for addition a+b=b+a Example: 5 + 7 = 7 + 5 12 = 12 2 BASIC LAW OF NATURAL NUMBERS Let a, b and c be any number. 3. Associative law for addition a + (b + c) = (a + b) + c Example: 12 + (13 + 22) = (12 + 13) + 22 12 + 35 = 25 + 22 47 = 47 4. Law of closure for multiplication a x b = ab Example: 2 x 4 = 2(4) 3 BASIC LAW OF NATURAL NUMBERS Let a, b and c be any number. 5. Commutative law for multiplication axb=bxa Example: 12 x -3 = -3 x 12 -36 = -36 6. Associative law of multiplication a (b x c) = (a x b) x c or a(bc) = (ab)c Example: 6 (-4 x -3) = (6 x -4) x -3 72 = 72 4 BASIC LAW OF NATURAL NUMBERS Let a, b and c be any number. 7. Distributive Law a(b + c) = ab + ac and or a(b – c) = ab – ac Example: 3(5 + 8) = 15 + 24 = 39 Example: 3(5 - 8) = 15 - 24 = -9 5 BASIC LAWS OF EQUALITY Let a, b and c be any number. 1. Reflexive property a=a 2. Symmetric property if a = b, then b = a 3. Transitive property if a = b and b = c, then a = c. - That is, things equal to the same thing are equal to each other 6 BASIC LAWS OF EQUALITY Let a, b and c be any number. 4. If a = b and c = d, then a + c = b + d. That is, if equals are added to equals, the results are equal. 5. If a = b and c = d, then ac = bd. That is, if equals are multiplied to equals, the results are equal. 7 INEQUALITY A statement that one quantity is greater than or less than another quantity. Symbols used in Inequalities a>b a is greater than b a 1 and x ≤ 5. It is a compound sentence whose conjunction is "and." It says that x takes on values that are greater than 1 and less than or equal to 5. 10 x. INEQUALITY COMPOUND INEQUALITY – “or” versus “and” x < 1 or x > 5 Here, the values of x are either less than 1 or greater than 5. It should be clear that x could not be a number that is less than 1 and greater than 5. There is no such number. 11 x. INEQUALITY CONTINUED INEQUALTY a 0. 14 x. INEQUALITY THEOREMS OF INEQUALITIES THEOREM 1: We may add the same number to both sides of an inequality, and the sense will not change. If a > b, then a +c > b + c by following the theorem that we can transpose ( a > b, a – b > 0) We can prove the by adding and subtracting c a–b+c–c>0 (a+c) – (b +c) > 0 a+c>b+c 15 x. INEQUALITY THEOREMS OF INEQUALITIES THEOREM 2: We may multiply both sides of an inequality by the same positive number, and the sense will not change. If a > b and c > 0 then ac > bc for example 8 > - 5, If we multiply both sides with 3, 8 (3) > -5 (3) 24 > - 15 16 x. INEQUALITY THEOREMS OF INEQUALITIES THEOREM 2: We may multiply both sides of an inequality by the same positive number, and the sense will not change. to prove: a >b a–b > 0 – lets multiply by c c( a – b) >0 ac – bc >0 ac > bc This theorem also allows us to divide both sides by the same positive number, because division is multiplication by the reciprocal. 17 x. INEQUALITY THEOREMS OF INEQUALITIES THEOREM 3: If we multiply both sides of an inequality by the same negative number, the sense of the inequality changes. If a > b and c < 0 then ac < bc for example -5 < 12, If we multiply both sides with -3, -5 (-3) > 12 (-3) 15 > - 36 18 x. INEQUALITY THEOREMS OF INEQUALITIES THEOREM 3: If we multiply both sides of an inequality by the same negative number, the sense of the inequality changes. to prove: a >b a – b > 0 – lets multiply by c If c is negative so by the rule of sign, c( a – b) < 0 ac – bc < 0 ac < bc This theorem also allows us to divide both sides by the same negative number, because division is multiplication by the reciprocal. 19 INEQUALITY THEOREMS OF INEQUALITIES THEOREM 4: If we change the signs on both sides of an inequality, then the sense of the inequality will change.. If a > b then -a < -b for example -6 < 12, Change the sign both sides 6 > -12 20 INEQUALITY THEOREMS OF INEQUALITIES THEOREM 5: If a, b are both positive or both negative, then on taking reciprocals, the sense of the inequality changes. 𝟏 𝟏 If a > b then < 𝒂 𝒃 for example 7 > 3 Then 1 1 < 7 3 21 INEQUALITY SOLVING INEQUALITIES A linear inequality has this standard form: ax + b < c When a is positive, then solving it is identical to solving an equation: ax < c –b 𝑐 −𝑏 x< 𝑎 The only difference between solving an inequality and solving an equation, is when we multiply or divide by a negative number, the sense must change. 22 INEQUALITY SOLVING INEQUALITIES Solve inequality for x. a. 4x + 7 < 31 b. -2x + 7 < 23 Solution: Solution: 4x + 7 < 31 -2x + 7 < 23 4x < 31 – 7 -2x < 23 – 7 4x < 24 -2x < 16 x< 6 x > -8 23 INEQUALITY SOLVING FOR COMPOUND INEQUALITIES A compound inequality represents two inequalities. Example 2: -3 < 4x + 9 < 7 -3 < 4x + 9 < 7 -- 3 -9 < 4x +9 –9 < 7 -9 -12 < 4x < -2 −𝟏𝟐 𝟒𝒙 −𝟐 < < 𝟒 𝟒 𝟒 𝟏 𝟏 -3