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

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School
University of Waterloo
Department
Mathematics
Course
MATH 135
Professor
Wentang Kuo
Semester
Fall

Description
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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