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# MATH 135 Algebra Notes I (Sept 9 - Sept 13 ): Proofs, Truth Tables, Sets

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University of Waterloo

Mathematics

MATH 135

Wentang Kuo

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MATH 135: Algebra for Honours Mathematics
Notes I
Michael Brock Li
Computational Mathematics IA
September MXIII
Concordia Cum Veritate
I: Proofs
i: Statement
The de▯nition of statement is a sentence which is either true or false.
Example I:
2 + 2 = 4
For the next following example, we cannot determine whether this is true or
false since the variable is not given.
Example II:
x > 0
ii: Open Sentence
It contains one or more variables, where each variable has values that come from
a designated set called the domain of the variable, and where the sentence is
either true or false.
iii: Implications
The most common type of statement to prove. It takes the form of
If A is true;then B is true:
It can be written as,
If A; then B:
1 or
A implies B:
or symbolically
A ) B
**Do note that implication takes form of a compound statement, which
consists of more than one statement.
If A; then B:
where A is a statement that can be true or false, and B is a statement that can
be true or false.
iv: Hypothesis and Conclusion
Statement A is hypothesis. Statement B is conclusion. You assume that A is
true, and use the assumption to prove that B is true. It is crucial to identify
which statement is the hypothesis, the conclusion, or proving an implication.
Example III:
If x is a positive real number, then 10x > 0.
We identify the hypothesis as x is a positive real number.
We identify the conclusion as log10x > 0.
However, let’s give a tricky example:
Example IV:
\ABC = \XY Z whenever 4ABC is similar to 4XY Z
We identify the hypothesis as 4ABC is similar to 4XY Z.
We identify the conclusion as \ABC = \XY Z.
v: Proposition
A proposition is a true statement that has been proved by a valid argument.
Remark: A theorem is a signi▯cant proposition. A lemma is a helper propo-
sition. A corollary that follows immediately from a theorem.
But do aware that there is axiom, which basically is a statement that is as-
sumed to be true, but does not contain a proof. Thus, choosing axioms has to
be done very carefully since we use them to derive propositions and theorems.
2 II: Truth Tables
Let’s take the following statement as an example:
If ajb and bjc; then ajc:
Allow X be the statement ajb and Y be the statement bjc and Z be the statement
ajc. Then this will become
X and Y implies Z
If X;Y and Z truth values are known, then we would be able to determine the
true value of the compound statement.
i: Not
The de▯nition of NOT A, is :A.
A :A
T F
F T
If the statement A is true, the statement of NOT A is false.
ii: And/Or
A B A ^ B
T T T
T F F
F T F
F F F
A B A _ B
T T T
T F T
F T T
F F F
iii: Implies
A B A ) B
T T T
T F F
F T T
F F T
3 The ▯rst row demonstrates a true hypothesis and true conclusion. Therefore,
you will get a true statement.
The second row demonstrates a true hypothesis and a false conclusion. There-
fore, it makes sense to have a false statement.
The third and fourth row is very obscure. If assumed a false hypothesis, then
how come it results a true statement? We can see the way that if one is allowed
to assume an hypothesis which is false, any conclusion can be derived.
iv: If and Only If
A B A () B
T T T
T F F
F T F
F F T
Example I:
We will construct a table for :(A _ B).
A B A _ B :(A _ B)
T T T F
T F T F
F T T F
F F F T
Example II:
We will construct a table for A ) (B _ C).
A B C B _ C A ) (B _ C)
T T T T T
T T F T T
T F T T T
T F F F F
F T T T T
F T F T T
F F T T T

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