MATH1030 Tutorial for Week 2
Advice.
• The material in this tutorial is about some rudiments of set langauge which will be most relevant to elementary linear
algebra.
(For a more comprehensive introduction, you are referred to any standard textbook on elementary linear algebra or
elementary calculus that includes some mathematical proofs. Such a textbook usually contains basic material on sets,
functions and relations in an introductory chapter or in an appendix.)
• Throughout this tutorial all numbers that appear are real numbers, and the word ‘number’ means ‘real number’.
• If you have never encountered the notions of belonging, set, element (of a set), predicate previously, the passages below
may serve as a very in-formal introduction:—
(I) The un-defined notion ‘belong to’ is to be understood on ‘heuristic terms’ (as in our daily experience in the use of
the phrase). It is left un-explained.
Through this un-defined notion, the words ‘set’ (or ‘collection’ or ‘aggregation’), ‘element’ to be introduced.
Given any objects x, A, we agree that the three sentences below mean the same thing, and is regarded as the same
mathematical statement:—
(I.1) ‘x belongs to A.’
(I.2) ‘The object x is an element of the set A.’
(I.3) ‘The set A contains the object x as an element.’
This statement is provided with a short-hand in this chain of symbols:—
‘x ∈ A.’
We write x ∈
/ A exactly when the object x does not belong to the set A.
(II) A chain of words and symbols is referred to as a predicate with unknowns x1 , x2 , · · · exactly when the replacement
of all the unknowns by ‘concrete objects’ (such as, numbers, vectors, matrices) will result in a statement (for which
it makes sense to say whether it is true or false).
Remarks.
Every equation (or system of equations) you have encountered in school mathematics, and every
equation that you will encounter is a predicate. The unknowns in such an equation (or such a system of equations)
are the unknowns in the predicate.
But there are predicates which are not equations and are not about numbers (or vectors or matrices).
• The symbol R stands for the set of all real numbers.
• We introduce, through some examples, how the method of specification (or the set-builder method) will be used in this
course.
For simplicity, these examples are ones which are immediately accessible from school mathematics.
However, the principles behind are not restricted to these examples. They will be applied elsewhere in this course.
(1) Suppose B is a set, and Q(x) is a predicate with unknown x. Then the chain of symbols
{
}
u ∈ B ‘Q(u)’ is a true statement
stands for the set of those and only those objects which belong to B and which, upon substitution into
the unknown x in Q(x), yield true statements.
(2) Suppose P (x1 , x2 ) stands for an equation with finitely many unknowns x1 , x2 , concerned with real numbers (and
understood as such).
For each concrete real number s1 , s2 , denote by P (s1 , s2 ) the statement (which may be true or false) resultant by
the substitution ‘x1 = s1 , x2 = s2 ’ into the equation P (x1 , x2 ).
Then the chain of symbols
{
(s1 , s2 ) The equality ‘P (s1 , s2 )’ is a true statement.
}
stands for the set of all solutions of the equation P (x1 , x2 ) (or simply, the solution set of the equation
P (x1 , x2 )).
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(3) Suppose T, S are two given sets, and f (x) stands for an expression involving the symbol x.
(So ‘y = f (x)’ stands for a predicate with two unknowns, namely, y, x.)
Then the chain of symbols
{
y∈T
There exists some r ∈ S
such that the equality ‘y = f (r)’ holds.
}
stands for the set of those and only those objects which belong to T and which can be expressed in
the form f (r) under the condition that r is something amongst the elements of S.
Very often such a set is expressed (in short-hand) as
{
f (r) f (r) ∈ T and r ∈ S
}
.
If it is known that f (u) ∈ T whenever u ∈ S, then we simply write the set concerned as
{
f (r) r ∈ S
}
.
Expressing the set concerned in this way, we may think of it as the collection of all possible quantities that can be
‘generated’ using the expression f (x), with x ‘running through’ all possible elements of S.
Remark.
Dependent on the style preferred by a writer, the vertical stroke ‘
{
so-and-so
so-and-so
’ inside the expression
}
may be replaced by the colon ‘:’. However, the use of of the curly brackets (as ‘signposts’) is not negotiable.
• We introduce the notions of set equality in terms of belonging, set:—
Suppose S, T are sets.
Then we say S is equal to T as sets if the statements (†), (‡) holds:—
(†) For any object x, if x ∈ S then x ∈ T .
(‡) For any object y, if y ∈ T then y ∈ S.
In this situation, we agree to write S = T .
Remarks.
In plain words (and in daily language):—
∗ (†) reads: ‘Every element of S belongs to T (as well).’
∗ (‡) reads: ‘Every element of T belongs to S (as well)’
1. This question is about syntax in set notations.
(a) Which of the chains of symbols below are definitely explicit expressions (formed with the method of specification) of
the solution set of the equation 9x1 2 + 4x2 2 = 36 with unknowns x1 , x2 ? (There is no need to explain your answer.)
(I)
{
}
(V)
.
)
(s1 , s2 ) 9s1 2 + 4s2 2 = 36 .
{
}
(s1 , s2 ) : 9s1 2 + 4s2 2 = 36 .
(III)
}
{
(u, v) 9u2 + 4v 2 = 36 .
(IV)
(II)
(
(s1 , s2 ) 9s1 2 + 4s2 2 = 36
{
(s, t) 9u2 + 4v 2 = 36
}
.
(s1 , s2 ) : 9s1 2 + 4s2 2 = 36 .
{
}
9u2 + 4v 2 = 36 (u, v) .
(VII)
{
}
(u, v), 9u2 + 4v 2 = 36 .
(VIII)
(VI)
(b) Here a, b stand for three fixed real numbers, and T, S stand for the set of all positive integers, the set of all integers
respectively.
Which of the chains of symbols below are definitely explicit expressions (formed with the method of specification) of
the set of those and only those positive integers of the form a + kb under the condition that k is an integer? (There
is no need to explain your answer.)
2
{
(I)
y∈T
There exists some r ∈ S
such that y = a + rb.
x∈T
There exists some r ∈ S
such that x = a + rb.
{
(II)
{
(III)
(V)
{
}
a + rb ∈ T and r ∈ S .
}
{
r ∈ S a + rb ∈ T .
(VII)
{
}
x ∈ T a + rb, r ∈ S .
(VIII)
{
}
x x = a + rb .
(IX)
{
}
x ∈ T x = a + rb for any r ∈ S .
(X)
{
}
a + rb .
(XI)
{
}
a + rb, r ∈ S .
(XII)
(VI)
.
}
.
}
x∈T
There exists some r ∈ S
such that y = a + rb.
z∈T
There exists some u ∈ S
such that z = a + ub.
x∈T
x = a + rb for some r ∈ S.
{
(IV)
}
.
}
.
}
.
{
a + rb
2. This question is about the notion of belonging, especially in the context of sets formed by the method of specification.
For the object x and the set A as specified in each part below, decide whether it is true that x ∈ A. Justify your answer.
}
u ∈ R u3 − 3u − 2 = 0 .
{
}
(b) x = (2, 1), A = (u, v) 3u2 − 4v 2 = 6 .
{
}
There exists some integer s
(c) x = 9, A =
v∈R
.
such that v = 4s + 1
{
}
s∈R
(d) x = 7, A = s4 + s2 + 1
.
{
}
There exists some integer s
(e) x = 4, A =
v∈R
.
such that v = 3s + 2
}
{
s2 − 3 ∈ R and s ∈ R
.
(f) x = −5, A = s2 − 3
(a) x = 2, A =
{
3. This question is about the notion of set equality, especially on the logical framework of an argument for a set equality.
For the sets A, B as specified in each part below, prove that A = B, with direct reference to the definition of set equality:—
{
}
(a) A = (x1 , x2 ) x1 ∈ R and x2 ∈ R and 2x1 + 3x2 = 7 ,


x1 ∈ R and x2 ∈ R and


there exist some s ∈ R
B=
(x1 , x2 )
.


such that x1 = 2 − 3s and x2 = 1 + 2s
{
}
(b) A = (x1 , x2 ) x1 ∈ R and x2 ∈ R and 4x1 = x2 2 ,


x1 ∈ R and x2 ∈ R and


there exist some t ∈ R
B=
(x1 , x2 )
.


such that x1 = t2 and x2 = 2t
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