Lec1.1 - L1.1 Lecture Notes: Logic Justification: Precise...

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L1.1 Lecture Notes: Logic Justification: Precise and structured reasoning is needed in all sciences including computer science. Logic is the basis of all reasoning. Computer programs are similar to logical proofs. Just as positive whole numbers are the fundamental units for arithmetic, propsitions are the fundamental units of logic. Proposition:  A statement that is either true or false. E.g. Today is Monday Today is Tuesday The square root of 4 is 2 The square root of 4 is 1 2 is even, and the square of two is even, and 3 is odd and the square of 3 is odd. The Panthers can clinch a playoff berth with a win, plus a loss by the Rams, a loss or tie by the Saints and Bears, a win by the Seahawks and a tie between the Redskins and Cowboys. (Copied verbatim from the sports page 12/26/2004.) Propositions may be true or false and no preference is given one way or the other. This is sometimes difficult to grasp as we have a “natural” preference for true statements. But “snow is chartreuse” and “snow is white” are both propositions of equal standing though one is true and the other false. Non-propositions: What is today? Is today Monday? Questions are not propositions. You can’t judge whether the question itself is true or false, even though the answer to the question may be true or false. Show me some ID! Similarly, imperative statements lack a truth value. 2x=4 x=y Statements with undetermined variables do not have truth value 1
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xy=yx looks like a true proposition but is it multiplication or concatenation? If you don’t know, it’s not a proposition. This sentence is false (paradoxes are not propositions) Propositions may be abbreviated by letters: p,q,r, etc. Thus logic may sometimes be called symbolic logic because we use symbols like p or q to stand for propositions. E.g., let p be the proposition “Today is Monday.” Combining Propositions . Simple or atomic propositions may be combined into compound propositions by use of logical operators . Thus if p is a proposition, then “it is not the case that p” is also a proposition and is denoted as ¬ p (“not p”) and has the opposite truth value. E.g. If “Today is Friday” = p then “Today is not Friday” is denoted by ¬ p. Not-p is a compound proposition. Clearly, if p is T, then ¬ p is F and if p is F then ¬ p is T. This may be expressed concisely in a TRUTH TABLE: A truth table must take all possibilities into account. Here there are two. p ¬ p T F F T This is similar to the negation operator in arithmetic or algebra, but logic is much simpler. Its variables have only two possible value as opposed to the infinitely many values of arithmetic. Consider the statement:
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Lec1.1 - L1.1 Lecture Notes: Logic Justification: Precise...

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