ࡱ> f%` bjbjNN 8",,gK(  ooo8(pqL sx(xx@y$hf ŕT x@y^>@xp @yGl @yr ="o/*t0tݨtTt Le6,Ǡ*^   eo$   o    SM242 Week 1 (Notes based on Epp Text, Chapter 1) I. Logic Are there "rules" or "laws" that govern proper reasoning? Can we prove, mathematically, that a particular argument is valid? To what extent can reasoning be automated? 1. Statement (proposition): a sentence that is either true or false, but not both. Example. Determine if each of the following sentences is a statement. a. 5 + 3 = 7 21 / 7 = 3 c. MIDN Miller is President of the United States d. Ah,the joy of SM242! e. He is a midshipman MIDN Avworo has written C++ programs. Baltimore is the capital of Maryland. h. 3 * z < 9 i. Every even integer greater than 2 is the sum of two primes j. This sentence is false. 2. Compound Statements: a more complex statement composed of simpler statements. The truth-value of a compound statement depends on the truth values of the simpler component statements. The component statements in a compound statement are often referred to as statement variables. A compound statement usually consists of statement variables joined by logical connectives. 3. Logical Connectives: Not (negation) Mathematical symbol: ~ ~p means "it is not the case that p" where p is some statement. Other notations:  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  C++ notation: The negation of a statement has the opposite truth-value from the statement. If p is true, ~p is false. The definition of negation is displayed in a truth table  p ~p  F T T F And (conjunction) Mathematical symbol:  EMBED Equation.DSMT4   EMBED Equation.DSMT4  means it is the case that p and q are true. C++ notation: The conjunction of two statements p and q is true when and only when both p and q are both true. The definition of conjunction is displayed in a truth table  p q  EMBED Equation.DSMT4   F F F T T F T T Example p: MIDN Berrios is an officer q: MIDN Berrios is a gentleman  EMBED Equation.DSMT4 : Or (disjunction) Mathematical Symbol:  EMBED Equation.DSMT4  Also called "inclusive or" C++ notation: The disjunction of two statements  EMBED Equation.DSMT4  is true if either p or q is true, or if both p and q are true. The definition of disjunction is displayed in a truth table  p q  EMBED Equation.DSMT4   Example p: it is rainy q: it is cold  EMBED Equation.DSMT4 : Exclusive or Mathematical Symbol:  EMBED Equation.DSMT4  , XOR  EMBED Equation.DSMT4  is true if either p is true or q is true, but not both. The definition of exclusive or is displayed in a truth table  p q  EMBED Equation.DSMT4   Precedence of logical connectives Negation has higher precedence than conjunction( EMBED Equation.DSMT4 ) and disjunction( EMBED Equation.DSMT4 ). Conjunction and disjunction are usually considered to have the same precedence. Note that this differs from the C++ high-level language, in which conjunction has a higher precedence than disjunction. Example: ~ p  EMBED Equation.DSMT4  q is performed as Parenthesis can be used to override normal precedence. Example: It is important that we use parenthesis to avoid ambiguity Example: Consider p  EMBED Equation.DSMT4  q  EMBED Equation.DSMT4  r . Is this (p  EMBED Equation.DSMT4  q)  EMBED Equation.DSMT4  r or p  EMBED Equation.DSMT4  (q  EMBED Equation.DSMT4  r) ? 4. Translating English into symbolic logic statements Example Let p = It is chilly and q = It is overcast . Translate each of the following compound statements into a mathematical statement using symbolic logic. It is not chilly but it is overcast It is not both chilly and overcast It is neither chilly nor overcast Example Let p = 0 < x , q = x < 2 and r = x = 2. Translate each of the following compound statements into a mathematical statement using symbolic logic, given that x is a particular real number. x  EMBED Equation.DSMT4  2 0 < x < 2 0 < x  EMBED Equation.DSMT4  3 5. Assigning truth values to compound statements We know the truth table for p  EMBED Equation.DSMT4  q and s  EMBED Equation.DSMT4  r. But what is the truth table for a compound statement, such as ~(p  EMBED Equation.DSMT4  q)  EMBED Equation.DSMT4  (s  EMBED Equation.DSMT4  r)? To develop the truth table for a complex statement we must consider each combination of truth values for the component statements. Example: Write the truth table for ~ p  EMBED Equation.DSMT4  q Start by making columns for p and q, considering all possible combinations of true and false.  p q  Now, use the definition of ~ to write a column for ~p. Then, write a column for ~ EMBED Equation.DSMT4  using the definition of disjunction.  p q ~ p ~ EMBED Equation.DSMT4   Example Write a truth table for (p  EMBED Equation.DSMT4  q)  EMBED Equation.DSMT4  ~(p  EMBED Equation.DSMT4  q)  p q  Have you seen this truth table before? Example Write a truth table for ( p  EMBED Equation.DSMT4  q )  EMBED Equation.DSMT4 ~ r   6. Logical Equivalence Two (compound) expressions are logically equivalent if and only if they have identical truth values for all possible combinations of truth values for the sub-expressions. If A and B are logically equivalent, we write  EMBED Equation.DSMT4 . (Another notation for logical equivalence is  EMBED Equation.DSMT4 ; that is, if A and B are logically equivalent then  EMBED Equation.DSMT4 .) To test if A and B and logically equivalent, construct truth tables for A and B. If the truth values of A and B match for all rows in the truth table, then  EMBED Equation.DSMT4 . Example: Are p and ~ (~ p) logically equivalent?  p ~ p ~ (~ p)  F T F T F T   Example: Are ~  EMBED Equation.DSMT4  and ~ EMBED Equation.DSMT4  ~q logically equivalent? p q  EMBED Equation.DSMT4  ~ EMBED Equation.DSMT4  ~ EMBED Equation.DSMT4  ~ EMBED Equation.DSMT4  ~ EMBED Equation.DSMT4  ~q  F F F T T T T F T F T T F F T F F T F T F T T T F F F F   7. DeMorgans Laws Example. Show that ~( p  EMBED Equation.DSMT4  q)  EMBED Equation.DSMT4 ~p  EMBED Equation.DSMT4  ~q   Think about this equivalence. Does it make sense? Consider the compound statement: MIDN Mincks is 20 years old and he is in the Navy. For this to be true, it must be the case that it is true that MIDN Mincks is 20 years old and it is true that MIDN Mincks is in the Navy. Now think of the negation of this compound statement. For this to be false, one or both of the parts of the compound statement must be false. So, the negation of MIDN Mincks is 20 years old and he is in the Navy is We can also slow that ~(  EMBED Equation.DSMT4  q )  EMBED Equation.DSMT4 ~ p  EMBED Equation.DSMT4  ~ q  DeMorgans Laws Example: What is the negation of the statement: It is cold or it is raining. 8. Tautologies A tautology is a statement that is always true. Example: Consider  EMBED Equation.DSMT4 ~ p  p ~ p  EMBED Equation.DSMT4 ~ p  F T T T F T 9. Contradictions A contradiction is a statement that is always false. Example: Consider  EMBED Equation.DSMT4  ~ p  p ~ p  EMBED Equation.DSMT4  ~ p F T F  T F F 10. Summary of Logical Equivalences (Text Chapter 1 page 14) 11. Using logical equivalences to simplify statements We can use the table on page 14 of the text to simplify statements. Example. Simplify the following compound statement: ( p  EMBED Equation.DSMT4  ~q )  EMBED Equation.DSMT4  [ ( p  EMBED Equation.DSMT4  q )  EMBED Equation.DSMT4  q ] Applying the Absorption Law to the right side (the side in square brackets), we have: ( p  EMBED Equation.DSMT4  ~q )  EMBED Equation.DSMT4  q Then applying the Distributive Law, we have: ( p  EMBED Equation.DSMT4  q )  EMBED Equation.DSMT4  ( ~q  EMBED Equation.DSMT4  q ) Then, using the Negation Law, we have: ( p  EMBED Equation.DSMT4  q )  EMBED Equation.DSMT4  T where T is true. Then, using the Identity Law, we finally have: ( p  EMBED Equation.DSMT4  q ) or, since parenthesis arent needed, p  EMBED Equation.DSMT4  q. Example. Simplify the following compound statement: ~(~p  EMBED Equation.DSMT4  q)  EMBED Equation.DSMT4  12. Implication We want to be able to reason from a hypothesis to a conclusion. We want to be able to say If p is true, then q must be true. or, more compactly if p, then q. 8 9 This is written symbolically as pgq. This is also called a  conditional. A. Examples: If MIDN Collard is a midshipman then MIDN Collard is in the Navy. If x > 2 then x > 0. If snow accumulation exceeds 50 inches then classes are canceled. Truth table for pgq.  p q pgq F F F T T F T T Suppose I tell you,  If you have a 95 average then you will get an A.  hypothesis conclusion When is this sentence false? Answer:  p q pgq F F F T T F T T If we have statements p and q, then pgq is read  p implies q and means that pgq is true. C. Precedence: In expressions that contain  g , the  g is evaluated last. D. English expressions which mean g In addition to expressing pgq as  if p then q, the following English expressions are also used (note that these expressions parallel the normal English meaning of if p then q): if p, q p implies q q if p q whenever p E. Example. Construct a truth table for the statement p  EMBED Equation.DSMT4  ~ q g ~ p   F. Example: Show that the conditional pgq is logically equivalent to that statement ~ p  EMBED Equation.DSMT4  q. That is, show pgq  EMBED Equation.COEE2  ~ p  EMBED Equation.DSMT4  q   G. Example. Prove that ~( pgq ) is logically equivalent to p  EMBED Equation.DSMT4  ~q. (Do on your own!) 13. The contrapositive: The contrapositive of pgq is ~ q g~ p. A conditional statement is logically equivalent to its contrapositive.   Examples: Write each of the following in its equivalent contrapositive form. If today is Wednesday then tomorrow is Thursday. If the temperature goes up then it will be hotter. If today is Friday then I have a quiz today Suppose we have the conditional p g q . The contrapositive is ~ q g ~ p which, in English, is Think about this sentence. It says if q does not occur, then p does not occur. This means that p can only occur if q occurs. Or, to put it more succinctly: To summarize, the following are all equivalent: if p then q if p, q q if p q whenever p p implies q p g q if not q then not p ~ q g ~ p p only if q 14. Example Let a =  You can access the Internet from Bancroft Hall c =  You are a computer science major f = You are a plebe Translate the following English sentence to a logical expression: You can access the Internet from Bancroft Hall only if you are a computer science major or you are not a plebe. 15. Necessary and sufficient conditions Consider the sentence p is a sufficient condition for q. Isnt this what we normally mean in English when we say if p then q? Hopefully you answered yes, and so we have yet another meaning for p g q, namely:  p is a sufficient condition for q. Try to hang on for one more! Consider the sentence  q is a necessary condition for p. To say that q is a necessary condition for p means the occurrence of q is necessary for p to occur; if q does not occur then p can t occur: ~ q g ~ p (if not q, then not p). But considering the contrapositive, this is the same as p g q (i.e., if p then q). So, to summarize again, the following are all equivalent: if p then q if p, q q if p q whenever p p implies q p g q if not q then not p ~ q g~ p p only if q p is a sufficient condition for q q is a necessary condition for p 16. Review Question 1. Consider the sentence: MIDN Craner will get an A only if he hands in all his homework. (a) Express this as if A then B. (b) What would be an alternative equivalent way to express this? (c) Is this a third way to express the original sentence? If MIDN Craner handed in all his homework, then he received an A. (d) Express the original sentence as Blah is a necessary condition for Bleh. (e) Express the original sentence as Blah is a sufficient condition for Bleh. (f) Given the original sentence, can I conclude that MIDN Craner having handed in all his homework is a sufficient condition for his having received an A? Question 2. Consider the sentence: If MIDN OBrien is a midshipman at the USNA, then he is not married. (a) Express this as Blah is a necessary condition for Bleh. (b) Is this correct: Being a midshipman at USNA is a necessary condition for MIDN OBrien to be unmarried. (c) Express the original sentence as Blah is a sufficient condition for Bleh. (d) From the original sentence, is it correct to conclude: Being unmarried is a sufficient condition for me to conclude that MIDN OBrien is a USNA midshipman. (e) Express the original sentence as p only if q. Question 3. Consider the sentence: The fact that MIDN Cunha lives in Bancroft Hall is a sufficient condition for me to conclude that he is a midshipman. (a) Convert this statement to if p then q: (b) Express this as Blah is a necessary condition for Bleh. (c) From the original sentence, is it correct to conclude that living in Bancroft Hall is a necessary condition for being a midshipman? (d) Express the original sentence as p only if q. Question 4. Consider the sentence: MIDN Harrisons being a senior is a necessary condition for him to be Brigade Commander. (a) Convert this statement to if p then q: (b) Convert this statement to an alternative form of if p then q: (c) Express the original sentence as Blah is a sufficient condition for Bleh. Question 5. What is the truth value of this statement: If 2 + 2 = 5 then MIDN Tublin is a deity. Question 6. What is the negation of the following statement: If MIDN Hess is a senior then she is cynical. Question 7. Suppose p and q are statements and p ! q is known to be false. What is the value of p  EMBED Equation.DSMT4  q? Question 8. What is the negation of the following statement: If n is prime then n is odd or n is two. Question 9. Suppose Midshipman Regulations are modified such that midshipmen can own pets (and harbor them in their Bancroft Hall room). However, Regulation 143.46.51 Section 15, Paragraph 5, Subparagraph 3, Subsection 400, says: It is against regulation for any midshipman to keep more than 2 dogs and 2 cats in his room in Bancroft Hall. 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Is he in violation of the aforementioned regulation? Explain. Question 10. What is the negation of the statement x < 5 or x is odd ? Question 11. Your Company Officer tells you that he will make you Company Commander only if by senior year you have a CQPR greater than 2.8 and a military QPR greater than 3.0. When you reach senior year your CQPR is 3.5 and your military QPR is 3.6 but your Company Officer does not make you Company Commander. Did your Company Officer lie to you? Explain. 17. The converse: The converse of p ! q is q ! p. In other words, the converse of if p then q is if q then p. A. The converse of a statement is not logically equivalent to the statement. If a conditional statement is true, its converse may or may not be true. B. Examples: What is the converse of the following statements? If today is Wednesday then tomorrow is Thursday. If today is Labor Day, then today is a holiday. If I can run 2 miles, then I can run 1 mile. If I turned on the light switch, then the light came on. 18. The inverse: The inverse of p g q is ~ p g ~ q. In other words, the inverse of if p then q is if ~p then ~q. The inverse of a statement is not logically equivalent to the statement. If a conditional statement is true, its inverse may or may not be true. Examples. What is the inverse of the statements in the last example? C. Answer the following three True/False questions. i) A conditional statement and its converse are logically equivalent. ii) A conditional statement and its inverse are logically equivalent. iii) The converse and the inverse of a conditional statement are logically equivalent to each other. D. Example. For each of the following statements, write the contrapositive, converse and inverse If he is an Ensign, then he is a naval officer. Contrapositive: Converse: Inverse: If n is divisible by 10, then n is divisible by 5 and n is divisible by 2 Contrapositive: If n is not divisible by 5 or n is not divisible by 2, then n is not divisible by 10. Converse: If n is divisible by 5 and n is divisible by 2 then n is divisible by 10. Inverse: If n is not divisible by 10 then n is not divisible by 5 or n is not divisible by 2. 19. The biconditional Given two statements p and q, the biconditional p ! q, read  p if and only if q, is the statement that is true if p and q have the same truth values and is false if p and q have opposite truth values.  p q p ! q  Thus, the statement  p if and only if q is really saying two things: So saying  p if and only q is saying: or, symbolically B. Again, p ! q is logically equivalent to the conjunction of p g q and q g p. 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The following is an example of a biconditional: An integer is even if and only if it is divisible by 2. D. To summarize, the following are all equivalent p ! q ( p g q )  EMBED Equation.DSMT4  ( q g p ) p is a necessary and sufficient condition for q p if and only if q E. Example: Express the statement  I am happy if an        T V X Z 46 $ & F 8p8^8`a$gd4 $h^ha$gd4 $^a$gd4$a$gd4$^`a$gd4$^`a$gd4 hjln<DPQRTYah_hM8hYh45hYh46h9Lh4>*hfZbh)h4CJaJU h4>*hXG hXGh4j"h?`h4EHUjZL h4CJUVaJjh4Uh4OJQJh?`h46h42 lPQRSTUh^hgdL h^`hgdXG  ^gd4 gd4  !gd4 ^`gd4 & F 8p8^8`gd4d only if I have found true love as the conjunction of two if-then statements. F. Notation. From now on, we will very often abbreviate if and only if as iff.     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