ENGR121 Engineering Mathematics FoundationsLecture Notes

Mark McGuinnessSMS, Victoria University of Wellington

March 5, 2020

These are lecture notes for ENGR 121. References are made to the recommended text,“Engineering Mathematics: A Foundation for Electronic, Electrical, Communications andSytems Engineers” (4th Edition), by Anthony Croft, Robert Davison, Martin Hargreavesand James Flint, Pearson, 2012, which is followed very closely in this course.

I will at the beginning of each week, put onto the course website the notes for the materialto be covered that week.

Lectures are also planned to be recorded on video, so that students can review any materialthey wish. The link to recorded lectures will be from Blackboard: click on ENGR121,then on “EchoCenter” in the left frame. This is still being set up; I hope it will be workingvery soon.

Week One.

1 Set Theory

(Croft, section 5.2)

A set is any collection of objects, things or states. Examples include

A = {1, 0}

B = {off, on}

D = {0,1,2,3,4,5,6,7,8,9}

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The set of icons

Set Membership:

The symbol ∈ means “is a member of ”. For example

5 ∈ D

reads “Five is a member of the set D” .

The symbol /∈ means “is not a member of”. For example

5 /∈ A

reads

“Five is not a member of the set A” .

1.1 Numbers

Some special sets of numbers are noted here:

Rational Numbers are defined to be of the form p/q, where p and q are integers, and q isnonzero. They can be expressed as decimal expansions. A decimal expansion (sometimessimply called a decimal) is where a number is written in base ten, with each digit indicating

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how many times a particular power of ten appears in a number, and with the decimalpoint just to the right of the 100 position. For example, 1.23 means 1× 100 + 2× 10−1 +3× 10−2.

A decimal fraction is a rational number where the denominator (the bottom line) is apower of ten, that is, p/q where q = 10m, m ∈ Z. All terminating decimal expansions aredecimal fractions. Decimal fractions are rational, but not all rationals can be written as adecimal fraction. For example, 1/3 when written as a decimal number never terminates,so cannot be written as a decimal fraction.

A rational number, when expressed as a decimal expansion, either terminates or repeatsindefinitely.

For example, this one terminates:

7

2=

35

10= 3.5 .

And this one repeats indefinitely the pair 63:

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11= 0.63636363 . . . = 0.63 .

Check out10

3= 3.33333 . . . = 3.3

Irrational numbers P cannot be written in the form p/q, where p and q are integers. Thisdefines them.

When written as a decimal expansion, an irrational number is infinitely long and neverrepeats or terminates. Examples of irrational numbers are π ≈ 3.1415927 . . .,

√2 ≈

1.414 . . .. The decimal digits never form a repeating sequence.

The set of real numbers, R, which loosely speaking contains all of the possible values takenby distances along a line (the number line), consists of the union of the set of rationalnumbers with the set of irrational numbers.

1.2 Set Notation

Curly brackets are used to indicate there is a set inside them. The set of positive realnumbers can be denoted as

R+ = {x : x ∈ R and x > 0}

which reads R-plus is the set of numbers x, such that x is real, and x is greater thanzero.

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Two sets are equal if they contain exactly the same members.

The order in which we list the members or elements of a set does not matter.So {1, 2, 3} = {2, 3, 1}.

1.3 Venn Diagrams

Venn diagrams are a pictorial way to look at sets that helps with understanding. Thefigure below shows the set {1, 2, 3, 4} and the set {3, 4, 5}.

The universal set E contains all the elements of possible interest. In the figure below,this is (rather arbitrarily) the numbers {1, 2, 3, 4, 5, 6, 7, 8}

1.4 Intersection

The intersection of two sets A and B is a new set, which contains all of the elements thatare common to both A and B. The notation and mathematical definition are

A ∩B = {x : x ∈ A and x ∈ B} .

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The intersection of the sets {1, 2, 3, 4} and the set {3, 4, 5} is the set {3, 4}.

If the intersection of two sets is empty (has no elements) we say these sets are disjoint,and we can write

A ∩B = φ .

That is, the empty set is denoted φ. It can also be denoted { }.

1.5 Union

The union of two sets A and B is the set of all elements that are either in A or in B; thatis, the set of all elements of A and those of B. We write

A ∪B = {x : x ∈ A or x ∈ B}

The union of the set {1, 2, 3, 4} and the set {3, 4, 5} is

the set {1, 2, 3, 4, 5}.

1.6 Subset

If the elements of A are also members of the set B, we say that A is a subset of B, andwe can write or state that A ⊂ B.

We can define the concept of subset in a mathematically more rigorous way as

A ⊂ B iff ∀x ∈ A, x ∈ B .

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I would read this as A is a subset of B if and only if, for all elements x in A, x is (also)in B.

Note the use above of the for all symbol, ∀.

1.7 Set Complement

The complement A of the set A is defined as:

Given a universal set E and a set A in this universal set, we call the set of members of Ethat are not in A, the complement of A, and label it A.

Note that A ∪ A = E, A ∪ A = A, A ∪ E = E, A ∩ φ = φ, and A ∩ A = φ.

1.8 The Laws of Set Algebra

Most of these laws are obvious when you inspect the Venn diagrams. The laws in thefirst set would be proven in a more advanced discrete mathematics course, but not in thiscourse. You should be able to derive the second set of laws from the first set.

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2 Sets, Relations and Functions

We may want to relate the members of two sets, for example by saying

“square it, or square it and add one”,

giving a rule for going from a member in one set to one or more members in another set.This example is illustrated below in what is called a mapping diagram. We call such arule, a relation.

A mapping diagram. A map gets you from A to B.

A relation r is defined to be:

a rule that maps from every element in set A to one or more elements in set B, which weexpress mathematically as

r : A→ B .

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The above expression says “r is a relation that maps from set A (in)to set B”, and wecall A the domain (the set from which we choose our input) and B the co-domain (theset into which we map), of the relation r. Some elements of B might not be mapped to.We call the set of elements that are actually mapped to, the range. This will be a subsetof the co-domain1.

A function is a relation that maps from each element in the domain, to just one element inthe range. Note that each element means every element. More about functions soon.

3 Numbers

(Croft, section 2.2)Much of this has been covered earlier in these notes. Just two more small points toadd:

Factorial notation is a useful way to write descending products, for example

6! = (6)(5)(4)(3)(2)(1) = 6× 5× 4× 3× 2× 1 = 720 ,

and in general we define

n! = n(n− 1)(n− 2)(n− 3) . . . (5)(4)(3)(2)(1) .

A working definition of a real number is a number that represents distance along aninfinite line called the real line. Real numbers can be represented as points on the realline. They consist of all of the rational numbers, together with all of the irrationalnumbers. The set of real numbers is denoted R. A small section of the real line is calledan interval.

1Note that Croft defines domain as the set from which we choose our input, and he also says that arelation maps elements of A to elements of B. This suggests that all elements in the domain are mappedby the relation, but seems to allow the possibility that there are some elements in the domain which wedon’t know what happens to, which are not mapped. A relation is often defined as a set of ordered pairs(x, y), and a more common definition of domain found online is that the domain is the set of first terms xthat appear in the relation. We are taking the interpretation that Croft means the domain of a relationis only those elements which are mapped by the relation.

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4 Inequalities

(Croft, section 1.6)Inequalities are about order, whether some (real) number is bigger or smaller or equal toanother number. If a number a is to the right of a number b on the number line, we saythat a is greater than b. If a number a is to the left of a number b on the number line,we say that a is less than b.

a > b means a is greater than b.a < b means a is less than b.a > b means a is greater than or equal to b.a 6 b means a is less than or equal to b.

The signs > and < are called strict inequalities. The signs ≥ and ≤ are called weakinequalities.

Some care is needed in manipulating inequalities, especially when there is a sign change(from positive to negative or vice-versa). Rules are:

If a > b then you can add any real number to both sides, positive or negative. That is,it follows that a + k > b + k ,∀k ∈ R, and that a − k > b − k ,∀k ∈ R. Similarly, theother inequalities do not change when you add or subtract the same real number fromboth sides. Note that the sign of the real number being added (or subtracted) does notmatter.

When multiplying or dividing by a real number k,

it matters whether k is positive or negative.

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If k is positive, the inequality is unchanged, for example, if a > b, then ka > kb , and a/k >b/k , ∀k ∈ R+.If k is negative, the inequality is reversed, for example, if a > b, then ka < kb , and a/k <b/k , ∀k ∈ R−.

If k is zero and is multiplied through, both sides are then zero and strict inequalities failto hold. There is not much point in multiplying both sides by zero, in any case. And youdon’t ever want to divide by zero either.

Example: solve x2 > 4

Solution: If x2 > 4 then x2 − 4 > 0, subtracting 4 from both sides. Hence, factoring theleft-hand side, (x − 2)(x + 2) > 0. For a product to be positive, either both terms arepositive or both are negative. Hence,

eitherx− 2 > 0 AND x+ 2 > 0

which givesx > 2 AND x > −2

which gives x > 2 as the only way to satisfy both of these at once;

orx− 2 < 0 AND x+ 2 < 0

which givesx < 2 AND x < −2

which gives x < −2.

This illustrates a general rule:

If x2 > k then x >√k or x < −

√k .

If x2 < k then −√k < x <

√k .

5 Functions

(Croft, section 2.3)A less rigorous definition of a function than the previous one, is that a function is a rule,that when given an input, produces a single output. The more rigorous definition in termsof sets is:

A function is a relation which maps each element of a set D (its domain) to a singleelement of a set C (its co-domain).

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Note that the definitions of co-domain and range, from the section on relations, still applyfor functions.

For example, “triple the input”:f : x→ 3x

orf(x) = 3x

or simplyf = 3x .

Then examples of different possible inputs and outputs are

f(3) = 9

andf(t− 1) = 3t− 3 .

The input to a function is called the argument.

Example:

If f(x) = x2 write down

f(t); f(3); f(2x); f(−x); f(x+ 1); f(x−1).

Answers:f(t) = t2; f(3) = 9; f(2x) = 4x2; f(−x) = x2; f(x + 1) = (x + 1)2 = x2 + 2x + 1;f(x−1) = x−2.

The graph of a function is a plot of values of y = f(x) versus x, for some range of xvalues. x is called the independent variable, since you can choose it freely. y is calledthe dependent variable, since it depends on the input x.

If the domain of a function is not given explicitly, it is then taken to be the largest setpossible. For example, the function

g(x) = x2 − 2

has domain (−∞,∞), and range [−2,∞).

5.1 One-to-Many

Relations may be one to many, but functions may not. For example, the relation

x→ ±√x

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Figure 1: A graph of y = 2x+ 1 versus x.

Figure 2: A graph of y = x2 − 2 versus x.

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is one-to-many, and is not a function. The relation ”take the positive square root” is afunction. Functions must pass the “vertical line test” — vertical lines intersect the graphof a function no more than once. Other the graph is not of a function.

5.2 Many-to-one, one-to-one

Functions can be many-to-one or one-to-one. For example, f(x) = x2 maps both 2 and−2 to the value 4, so it is many-to-one. A good test for many-to-one is the horizontalline test: if there is a horizontal line that intersects the graph at more than one place, thefunction is many-to-one.

A one-to-one function gives different outputs if the inputs are different. All of the hori-zontal lines that intersect its graph do so at only one place.

Figure 3: A graph of a many-to-one function (dashed curve) and a one-to-one function(solid curve) on the same plot.

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5.3 Parametrically defined functions

A function is often given as y = f(x), or just y(x). Then a graph can be formed byjoining the dots, with x and y coordinates equal to (x, y(x)) or (x, f(x)). It can be usefulto express both x and y in terms of a third variable (an independent variable) called aparameter. For example,

x = t2, y = sin t

which has a graph as plotted in Fig. (4). Is this in fact a function?

A better title for this subsection would be Parametrically defined curves, since the curvesyou obtain often are not functions (since they are often one-to-many).

Figure 4: A graph of a parametrically defined curve.

5.4 Function Composition

Consider the function f(x) = 3x2. We can break it down into two simpler functions,“square the input”, and then “triple the input”. If we square x, then triple the result,we get f . Note that the order matters: if we triple x then square the result, we get(3x)2 = 9x2, not f .

Mathematically, if h(x) = 3x and g(x) = x2, we are rewriting

f(x) = 3x2 = 3(g(x)) = h(g(x)) .

We say that f is the composition of the two functions h and g. h(x) = 3x, andg(x) = x2.

[Can you think of a way to reverse the composition and still get f correctly? That is, tosquare the input last and not first?]

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