MTH 101 Elementary Mathematics I: Algebra and Trigonometry

Course focus: This study material presents the core topics in elementary algebra and introductory mathematical reasoning for MTH 111. It is organized for revision, class study, and examination preparation.

Study Sections

1. Elementary Set Theory
2. Operations on Sets
3. Venn Diagrams, Ordered Pairs, and Cartesian Products
4. The Real Number System
5. Sequences and Progressions
6. Series
7. Mathematical Induction
8. Theory of Quadratic Equations
9. Factorials, Combinations, and the Binomial Theorem
10. Exam High-Yield Formula Sheet
11. Practice Questions for Revision
12. References

1. Elementary Set Theory

1.1 Meaning of a set

A set is a well-defined collection of distinct objects. The objects in a set are called elements or members.

Examples:

• A = {2, 4, 6, 8} is a set of four even numbers.
• B = {dog, book, 7, Mary} is also a set because the elements are clearly listed.
• C = {8, a, 8} is the same as {8, a} because repeated elements are counted only once.

Two basic rules govern ordinary sets:

1. Repetition does not change a set.
2. Order does not change a set.

1.2 Set notation

Common notation:

• Use capital letters such as A, B, C for sets.
• Use small letters such as x, y, z for elements.
• Write x ∈ A to mean x is an element of A.
• Write x ∉ A to mean x is not an element of A.
• Write ∅ for the empty set, the set containing no element.

A set may be described by listing its elements or by giving a defining property.

Listing method:

Set-builder method:

Read this as: A is the set of all x such that x is an odd natural number less than 8.

1.3 Common number sets

The most common number sets in this course are:

Important convention: Some textbooks include 0 in N, while others do not. In this document, N = {1, 2, 3, ...}. If 0 is included, use W for whole numbers.

1.4 Equality, finite sets, infinite sets, and singleton sets

Two sets are equal if they contain exactly the same elements. The order of writing does not matter.

Example:

A finite set has a fixed number of distinct elements. If A is finite, n(A) means the number of elements in A.

Examples:

• A = {2, 4, 6}; n(A) = 3.
• B = {7, 7, 17}; n(B) = 2 because 7 is counted once.
• Z is infinite because the integers continue without end.

A singleton set is a set containing exactly one element. For example, {a} is a singleton.

1.5 Subsets and proper subsets

A set A is a subset of a set B if every element of A is also an element of B. We write this in words as A is a subset of B.

Example:

Every set is a subset of itself, and the empty set is a subset of every set.

A proper subset of B is a subset of B that is not equal to B.

Example:

The proper subsets of {3, 4} are ∅, {3}, and {4}.

Exam warning: {} and {{}} are not the same. The first is the empty set. The second is a set with one element, and that element is the empty set.

1.6 Power set

The power set of A is the set of all subsets of A.

Example:

If a finite set has n elements, then the number of subsets is:

Example:

Key takeaways

• A set is a collection of distinct objects.
• Order and repetition do not matter in ordinary sets.
• The empty set is a subset of every set.
• A power set contains all subsets of a set.
• A set with n elements has 2ⁿ subsets.

2. Operations on Sets

2.1 Universal set

A universal set is the larger set under discussion. Complements are always taken relative to a chosen universal set.

Example:

2.2 Union

Notation in this document:

Example:

Repeated elements are written once.

2.3 Intersection

The intersection of A and B is the set of elements common to both A and B.

Example:

2.4 Difference of sets

Example:

In general, A - B is not equal to B - A.

2.5 Complement

Important laws:

• A ∪ A^c = U.
• A ∩ A^c = ∅.

Example:

2.6 Symmetric difference

The symmetric difference of A and B is the set of elements that belong to exactly one of the two sets.

Example:

2.7 Laws of set theory

2.8 Counting formula for two sets

For finite sets:

Example:

In a class of 40 students, 20 take Chemistry, 25 take French, and 8 take both. Then:

2.9 Counting formula for three sets

For finite sets A, B, and C:

The final addition is necessary because the elements common to all three sets are subtracted too many times.

Key takeaways

• Union means either or both.
• Intersection means common elements only.
• Difference means elements in the first set but not the second.
• Complement depends on the universal set.
• De Morgan's laws are high-yield exam laws.
• In counting problems, always subtract overlaps correctly.

3. Venn Diagrams, Ordered Pairs, and Cartesian Products

3.1 Venn diagrams

A Venn diagram is a visual representation of sets and their relationships. The rectangle usually represents the universal set, while circles represent subsets.

Use Venn diagrams when a problem involves:

• Two or three groups.
• Overlapping membership.
• Students taking subjects.
• Items with multiple defects.
• Survey results.

Exam method for two-set problems:

1. Put the intersection in the overlap first.
2. Subtract the overlap from each individual set.
3. Add all occupied regions to get the ∪.
4. Subtract the ∪ from the universal total to get neither.

Example:

A survey has 400 students. 100 are smokers, 150 chew gum, and 75 do both.

3.2 Ordered pairs

An ordered pair is a pair of elements written in a fixed order, such as (a, b). The first entry is the first component, and the second entry is the second component.

In general:

Equality of ordered pairs:

Example:

3.3 Cartesian product

The Cartesian product A × B is the set of all ordered pairs (a, b), where a is in A and b is in B.

Example:

Usually, A × B is not equal to B × A.

3.4 Number of elements in a Cartesian product

If A and B are finite sets, then:

Example:

Key takeaways

• Ordered pairs depend on order.
• Cartesian product means all possible ordered pairings.
• A × B and B × A are generally different.
• If A has m elements and B has n elements, then A × B has mn elements.

4. The Real Number System

4.1 Meaning of real numbers

A real number is any number that can be located on the number line. Real numbers include positive numbers, negative numbers, zero, fractions, terminating decimals, repeating decimals, and irrational numbers.

The real number system is organized as follows:

N is contained in Z, Z is contained in Q, and Q is contained in R.

Irrational numbers are also contained in R, but they are not contained in Q.

4.2 Natural numbers and whole numbers

Natural numbers are counting numbers:

Whole numbers include zero:

4.3 Integers

Integers are positive whole numbers, negative whole numbers, and zero:

Integers do not have fractional or decimal parts.

Examples of integers: -5, 0, 12.

4.4 Rational numbers

Examples:

• 3 = 3/1.
• -5/7 is rational.
• 0.75 = 3/4.
• 0.333... = 1/3.

All integers are rational numbers because every integer can be written over 1.

4.5 Why repeating decimals are rational

Multiply by 10:

Subtract:

Therefore, a repeating decimal can often be converted into a fraction and is rational.

4.6 Irrational numbers

Examples:

• √(2)
• √(3)
• π
• e

Important note: A fraction with denominator zero is not an irrational number; it is undefined.

4.7 Properties of real numbers

For real numbers a, b, and c:

4.8 Absolute value

The absolute value of a real number is its distance from zero on the number line.

Examples:

Distance interpretation:

If the distance between x and 3 is at most 2, write:

Key takeaways

• Real numbers are all numbers on the number line.
• Rational numbers are fractions or terminating/repeating decimals.
• Irrational numbers are non-terminating and non-repeating.
• Absolute value represents distance, so it is never negative.
• Know the real number properties by name and by formula.

5. Sequences and Progressions

5.1 Sequence

A sequence is an ordered list of numbers formed according to a rule. Each number in the list is called a term.

Notation:

A finite sequence has a last term. An infinite sequence continues without end.

Example:

Therefore, the first four terms are 2, 3, 4, 5.

5.2 Finding the general term

To find a general term, observe the pattern in the sequence.

Example 1:

Sequence: -1, 1, -1, 1, ...

The sign alternates, so:

Example 2:

Sequence: 1, 1/4, 1/9, 1/16, ...

The denominators are square numbers:

5.3 Arithmetic progression

An arithmetic progression, or A.P., is a sequence in which the difference between consecutive terms is constant. This constant is called the common difference.

If the first term is a and the common difference is d, then:

The nth term is:

Example:

Find the 5th term of 5, 9, 13, ...

5.4 Finding an A.P. from two terms

If the 7th term of an A.P. is 27 and the 12th term is 47, find the sequence.

Subtract the first equation from the second:

Thus:

The sequence is 3, 7, 11, 15, ...

5.5 Geometric progression

A geometric progression, or G.P., is a sequence in which the ratio of consecutive terms is constant. This constant is called the common ratio.

If the first term is a and the common ratio is r, then:

The nth term is:

Example:

Find the 6th term of 1, 1/2, 1/4, ...

5.6 Finding a G.P. from two terms

If the second term of a G.P. is 8/9 and the sixth term is 9/2, then:

Divide u₆ by u₂:

Therefore:

So there are two possible geometric progressions.

Key takeaways

• A sequence is ordered; a set is not necessarily ordered.
• In an A.P., subtract consecutive terms to get d.
• In a G.P., divide consecutive terms to get r.
• A.P. nth term: u_n = a + (n - 1)d.
• G.P. nth term: u_n = ar^{n - 1}.

6. Series

6.1 Meaning of a series

A series is the sum of the terms of a sequence.

If a sequence is u₁, u₂, u₃, ..., u_n, then the sum of the first n terms is:

Summation notation writes this compactly:

Example:

6.2 Sum of an arithmetic progression

For an A.P. with first term a, common difference d, and n terms:

If the last term is l, then:

Example:

Find the sum of the first 12 terms of 15, 12, 9, ...

6.3 Number of terms in an arithmetic series

How many terms of 3, 7, 11, ... must be added to give 300?

6.4 Sum of a geometric progression

For a G.P. with first term a, common ratio r, and n terms:

Equivalently:

Both forms are algebraically the same. Choose the form that keeps calculations simple.

If |r| < 1, the infinite geometric series has sum:

Example:

Find the sum of the first 10 terms of 8 + 4 + 2 + ...

Key takeaways

• A sequence lists terms; a series adds terms.
• For A.P. sums, know S_n = n/2[2a + (n - 1)d].
• For G.P. sums, know S_n = a(1 - rⁿ)/(1 - r), r ≠ 1.
• Infinite G.P. sums require |r| < 1.
• When solving for n, reject negative or fractional answers if n counts terms.

7. Mathematical Induction

7.1 Meaning of mathematical induction

Mathematical induction is a proof method used to show that a statement is true for all positive integers, or for all integers starting from a specified integer.

It works like a chain:

1. Prove the first case is true.
2. Prove that whenever one case is true, the next case must also be true.
3. Conclude that all cases are true.

7.2 Standard induction steps

To prove a statement P(n) for all positive integers n:

1. Base step: Show that P(1) is true.
2. Inductive hypothesis: Assume P(k) is true for some positive integer k.
3. Inductive step: Use the assumption P(k) to prove P(k + 1).
4. Conclusion: Therefore, P(n) is true for all positive integers n.

7.3 Example: Sum of the first n odd numbers

Prove that:

Base step:

Inductive hypothesis:

Inductive step:

Then:

1 + 3 + 5 + ... + (2k - 1) + (2k + 1)

Using the inductive hypothesis:

Conclusion:

By mathematical induction, the formula is true for all positive integers n.

7.4 Example: Sum of the first n natural numbers

Prove that:

Base step:

Inductive hypothesis:

Assume:

Inductive step:

1 + 2 + 3 + ... + k + (k + 1)

Key takeaways

• Do not skip the base step.
• Do not assume the result is true for all n; assume only that it is true for n = k.
• The inductive step must prove the k + 1 case.
• Induction is especially useful for sums, divisibility, inequalities, and formulas involving n.

8. Theory of Quadratic Equations

8.1 Definition

A quadratic equation is an equation that can be written in the standard form:

Here:

• a is the coefficient of x².
• b is the coefficient of x.
• c is the constant term.

8.2 Methods of solving quadratic equations

Common methods include:

1. Factorization.
2. Completing the square.
3. Quadratic formula.

8.3 Solving by factorization

If a quadratic expression can be written as a product of two linear factors, use the zero-product property.

Zero-product property:

Example:

Factor:

So:

8.4 Solving by completing the square

Example:

Move the constant term:

Take half of -6 and square it:

Add 9 to both sides:

Factor the left side:

Take square roots:

8.5 Quadratic formula

The discriminant helps determine the type of roots:

Example:

8.6 Sum and product of roots

If α and β are roots of:

then:

Example:

8.7 Forming a quadratic equation from roots

If the sum of roots is S and the product of roots is P, then the monic quadratic equation is:

Example:

8.8 Useful root identities

If α and β are roots, then:

Key takeaways

• Always put the quadratic in standard form before identifying a, b, and c.
• Factorization is fastest when the quadratic factors easily.
• The quadratic formula works for every quadratic equation.
• Sum of roots is -b/a; product of roots is c/a.
• Given sum S and product P, write x² - Sx + P = 0.

9. Factorials, Combinations, and the Binomial Theorem

9.1 Factorial

For a positive integer n, n factorial, written n!, is the product of the positive integers from n down to 1.

Also:

Examples:

A useful identity is:

9.2 Combinations

Examples:

Useful properties:

9.3 Binomial theorem

A binomial is an expression with two terms, such as a + b, x + 1, or 2x - 3.

For a non-negative integer n:

Expanded form:

9.4 Pattern in binomial expansion

For (a + b)ⁿ:

1. There are n + 1 terms.
2. The power of a decreases from n to 0.
3. The power of b increases from 0 to n.
4. In every term, the sum of the powers is n.
5. The coefficients are combinations: C(n, 0), C(n, 1), ..., C(n, n).

Example:

9.5 Expansion with a negative term

Example:

9.6 General term of a binomial expansion

The general term in (a + b)ⁿ is:

Example:

9.7 Finding a particular power of x

Example:

General term:

For the term in x²:

So the required term is the 6th term:

Key takeaways

• Factorial means product down to 1.
• 0! = 1 is essential for combination formulas.
• C(n, r) = n!/[r!(n - r)!].
• The binomial expansion of (a + b)ⁿ has n + 1 terms.
• The general term is T_r+1 = C(n, r)a^n-rb^r.
• To find a required power of x, form the general term and equate the exponent of x to the required power.

10. Exam High-Yield Formula Sheet

Set theory

Real numbers

Sequences and series

A.P. nth term:

A.P. sum:

G.P. nth term:

Finite G.P. sum:

Infinite G.P. sum:

Induction

1. Prove P(1).
2. Assume P(k).
3. Prove P(k + 1).
4. Conclude P(n) is true for all positive integers n.

Quadratic equations

Standard form:

Quadratic formula:

Sum of roots:

Product of roots:

Equation from sum S and product P:

Binomial theorem

11. Practice Questions for Revision

Set theory

1. If U = {1, 2, 3, ..., 10}, A = {1, 3, 5, 7}, and B = {3, 4, 5, 8}, find A ∪ B, A ∩ B, A - B, B - A, and A^c.
2. List all subsets of {a, b, c}.
3. How many subsets does a set with 7 elements have?
4. In a class of 60 students, 35 study Mathematics, 28 study Physics, and 12 study both. How many study neither?
5. Prove De Morgan's law for two small sets using a universal set of your choice.

Real numbers

1. Classify each number as natural, integer, rational, irrational, or real: -4, 0, 2/3, √(7), 0.252525..., π.
2. Convert 0.777... to a fraction.
3. Simplify |5 - 12| + |-3|.
4. Write in absolute value notation: the distance between x and -2 is less than 5.

Sequences and series

1. Find the first five terms of u_n = 3n - 2.
2. Find the general term of 2, 5, 8, 11, ...
3. Find the 20th term of the A.P. 7, 11, 15, ...
4. Find the sum of the first 15 terms of 4, 9, 14, ...
5. Find the 8th term of the G.P. 3, 6, 12, ...
6. Find the sum of the first 6 terms of 5, 10, 20, ...

Mathematical induction

1. Prove that 1 + 2 + 3 + ... + n = n(n + 1)/2.
2. Prove that 1 + 3 + 5 + ... + (2n - 1) = n².
3. Prove that 2 + 4 + 6 + ... + 2n = n(n + 1).
4. Prove that 1 + 5 + 5² + ... + 5^n-1 = (5ⁿ - 1)/4.

Quadratic equations

1. Solve x² - 5x + 6 = 0.
2. Solve 2x² + 3x - 2 = 0.
3. Solve x² - 4x - 1 = 0 by completing the square.
4. For 5x² - 7x + 2 = 0, find the sum and product of the roots.
5. Form a quadratic equation whose roots have sum -3 and product -10.

Binomial theorem

1. Evaluate 6!, 8!/6!, and C(8, 3).
2. Expand (x + y)⁵.
3. Expand (1 - 3x)⁴.
4. Find the 5th term in the expansion of (2x - 1)⁷.
5. Find the term independent of x in the expansion of (x² + 1/x)⁹.

12. References

OpenStax. (2021). Algebra and Trigonometry 2e. Rice University.

OpenStax. (2021). College Algebra 2e. Rice University.

Mathematics LibreTexts. Basic concepts of sets and set operations.

Mathematics LibreTexts. Mathematical induction: An introduction.

Akaligwo, E. C. (2020). Lecture Notes on Elementary Mathematics I (MTH 111): Algebra and Trigonometry.