Unit 2: Work, Exponents, and Set Theory

Unitary Method and Applications

Unitary Method

The unitary method is a technique for solving problems by first finding the value of a single unit, and then using that value to find the required value.

Example: If 10 pens cost $50, what is the cost of 7 pens?

1. Find the value of 1 unit:
   Cost of 10 pens = $50
   Cost of 1 pen = $50 / 10 = $5
2. Find the required value:
   Cost of 7 pens = 7 × (Cost of 1 pen) = 7 × $5 = $35

Problems on Time and Work

This is a common application of the unitary method, based on the concept of "work rate".

• Work Rate: The amount of work done in one unit of time.
• If a person 'A' can finish a job in x days, A's 1-day work rate is 1/x.
• If A and B work together, their combined rate is (A's rate) + (B's rate).

Example: A can do a job in 3 days. B can do the same job in 6 days. How long will it take them to do it together?

1. A's 1-day work rate = 1/3
2. B's 1-day work rate = 1/6
3. Combined 1-day rate = (1/3) + (1/6) = (2/6) + (1/6) = 3/6 = 1/2
4. If their combined rate is 1/2 of the job per day, they will finish the entire (1) job in 2 days. (Time = Total Work / Rate = 1 / (1/2) = 2).

Problems on Speed and Distance

Based on the fundamental relationship:

Formulas:
Distance = Speed × Time
Speed = Distance / Time
Time = Distance / Speed

Surds and Laws of Exponents

Surds

A surd is an irrational number expressed as a root of a rational number, such as √2, ∛5, etc. The expression cannot be simplified to a rational number.

Laws of Exponents (Indices)

For any real numbers 'a' and 'b' and rational numbers 'm' and 'n':

1. Product Rule: aᵐ × aⁿ = aᵐ⁺ⁿ
2. Quotient Rule: aᵐ / aⁿ = aᵐ⁻ⁿ (where a ≠ 0)
3. Power of a Power Rule: (aᵐ)ⁿ = aᵐⁿ
4. Power of a Product Rule: (ab)ⁿ = aⁿbⁿ
5. Power of a Quotient Rule: (a/b)ⁿ = aⁿ / bⁿ (where b ≠ 0)
6. Zero Exponent: a⁰ = 1 (where a ≠ 0)
7. Negative Exponent: a⁻ⁿ = 1 / aⁿ
8. Fractional Exponent: aᵐ/ⁿ = ⁿ√(aᵐ) (the n-th root of a to the power of m)

Introduction to Set Theory

Introduction to Sets

• Set: A well-defined collection of distinct objects. e.g., A = {1, 2, 3, 4}.
• Subset (⊂): A set A is a subset of set B if every element of A is also in B.
  • Example: If A = {1, 2} and B = {1, 2, 3}, then A ⊂ B.
• Empty Set (∅ or {}): A set with no elements. The empty set is a subset of every set.

Set Operations

• Union (A ∪ B): The set of all elements that are in A, or in B, or in both.
  • Example: {1, 2} ∪ {2, 3} = {1, 2, 3}
• Intersection (A ∩ B): The set of all elements that are in both A and B.
  • Example: {1, 2} ∩ {2, 3} = {2}
• Difference (A - B): The set of all elements that are in A but not in B.
  • Example: {1, 2} - {2, 3} = {1}
• Cartesian Product (A × B): The set of all possible ordered pairs (a, b) where 'a' is in A and 'b' is in B.
  • Example: {1, 2} × {a, b} = {(1, a), (1, b), (2, a), (2, b)}

Number of Elements of Sets

• n(A): Denotes the number of elements in set A.
• Number of Subsets: A set with 'n' elements has 2ⁿ subsets.
  • Example: A = {1, 2, 3}. n(A) = 3. Number of subsets = 2³ = 8.
• Principle of Inclusion-Exclusion:

  n(A ∪ B) = n(A) + n(B) - n(A ∩ B)