Discrete Mathematics Lecture # 6

Standard Connectives connective symbol and  or  not ~ if – then 

Non standard Connectives … but … … although … … or … but not both neither … nor … … if … … only if … … if and only if … … unless … … is necessary for … … is sufficient for … … if … otherwise … …unless…in which case

Conjunction There standard English expression for conjunction is AND but there are other numerous conjunction-like expressions as follows: But Yet Although Though Even though Moreover Furthermore However Whereas

Examples It is raining but I am happy. Although it is raining, I am happy. It is raining yet I am happy. It is raining and I am happy.

Disguised Conjunctions Other examples of disguised conjunctions involved relative pronoun are: Who Which That For example Jones is a former player who coaches basket ball. Jones is a former player and Jones coaches basket ball.

Sometimes ‘and’ is used as follows. keep trying, and you will succeed Odd use of ‘and’ Sometimes ‘and’ is used as follows. keep trying, and you will succeed keep it up buster, and I will clobber you give him an inch, and he will take a mile give me the money, and I’ll give you the product you do that, and I’ll kill you Truth table for (pq)  ~ (pq)

Negations For instance: It is not true that _________. It is false that _____________. It is not the case that ___________.

Exclusive OR Ram xor Sam will win (the election) Ram will win, or Sam will win, but not both at least one of them win ( R  S ) also & both of them will win…not! ~( R ^ S ) alternatively, Ram will win, and Sam will not win, ( R ^ ~ S ) or  Sam will win, and Ram will not win. ( S ^ ~R )

Neither…Nor it is neither raining nor sleeting it is not raining, and it is not sleeting ~ R ^ ~ S neither Jay nor Kay is sleeping Jay is not sleeping, and Kay is not sleeping ~ J ^ ~ K

Alternative Formulation it is neither raining nor sleeting it is not raining or sleeting it is not true that ( it is raining or it is sleeting ) ~ ( R  S )  ~ R ^ ~S

A  C Conditionals standard expression: if A then C which is symbolized: A  C A is the antecedent C is the consequent

A  C Variants of if…then if A, then C if A, C C if A all of these are symbolized the same: A  C ‘ if ’ always introduces the antecedent ‘ then ’ always introduces the consequent

all symbolized the same way Other variants of ‘if’ C provided A C in case A C supposing A provided A, C in case A, C supposing A, C all symbolized the same way A  C

Only if ‘ only if ’ is not equivalent to ‘ if ’ T F T F I will get an A only if I take all the exams versus I will get an A if I take all the exams T F I will get an A if I get a hundred versus I will get an A only if I get a hundred T F

‘only’ operates as a dual-negative modifier. How does only works? employees only authorized personnel only cars only right turn only ‘only’ operates as a dual-negative modifier. for example, (1) means to exclude anyone who is not an employee.

How does ‘only’ modifies ‘if’ by introducing two negations A only if B not A if not B if not B then not A ~ B  ~ A

Example A only if E not A if not E if not E then not A ~ E  ~ A I will get an A only if I take all the exams A only if E I will not get an A if I do not take all the exams not A if not E if I do not take all the exams then I will not get an A if not E then not A ~ E  ~ A

Unless unless = if not The following are all equivalent. I will pass only if I study P only if S I will not pass unless I study not P unless S I will not pass if I do not study not P if not S only if has two built-in negations unless has one built-in negation if has no built-in negation unless = if not

I will not pass unless I study Example I will not pass unless I study not P unless S not P if not S not P if not S if not S then not P ~ S  ~ P

I will play tennis if it is sunny; otherwise, I will play racket ball If…otherwise I will play tennis if it is sunny; otherwise, I will play racket ball T if S ; otherwise, R this answers two questions: what will I do IF it is sunny? play tennis what will I do OTHERWISE (i.e., IF it is NOT sunny)? play racket ball

If…otherwise if it’s sunny, then I’ll play tennis furthermore if it’s not sunny, then I’ll play racketball if S then T and if not S then R ( S  T )  (~ S  R )

Unless…in which case T unless R, in which case S I will play tennis unless it rains, in which case I will play squash T unless R, in which case S this answers two questions: what will I do unless it rains (i.e., if it does not rain)? play tennis what will I do in case it rains (i.e., if it does rain)? play squash

Unless…in which case if it does not rain, then I’ll play tennis furthermore if it does rain, then I’ll play squash if not-R then T and if R then S (~R  T )  ( R  S )

Necessary Conditions in order that I get an A it is necessary that I take 4 exams in order for me to get an A it is necessary for me to take 4 exams in order to get an A it is necessary to take 4 exams taking 4 exams is necessary for getting an A

Necessary Conditions if not A then not B ~A  ~B simplest paraphrase: A is necessary for B this amounts to saying if A does not happen, then neither does B if not A then not B ~A  ~B

Example if not E then not A ~E  ~A taking 4 exams E is necessary for getting an A A if E does not happen, then neither does A if not E then not A ~E  ~A

Sufficient Conditions in order that I get an A it is sufficient that I get a hundred in order for me to get an A it is sufficient for me to get a hundred in order to get an A it is sufficient to get a hundred getting a hundred is sufficient for getting an A

Sufficient Conditions simplest paraphrase: A is sufficient for B this amounts to saying if A does happen, then so does B if A then B A  B

Example getting a hundred H is sufficient for getting an A A if H does happen, then so does A if H then A H  A

Negation of Necessity ( ) if then not H is nec for A if not H then getting a hundred H is NOT necessary for getting an A A not H is nec for A if not H then not A ~ ( ~ H  ~ A ) ~ (~ H  ~ A ) it is not true that if you don’t get a H then you won’t get an A

Negation of Sufficiency Taking all the exams E is NOT sufficient for getting an A A not E is suf for A if E then A ~ ( E  A ) ~ (E  A ) it is not true that if you (merely) take the exams then you will get an A

Basic Statements A is necessary for B A is sufficient for B A is not necessary for B A is not sufficient for B

Combinations A is both necessary and sufficient for B A is necessary, but not sufficient, for B A is sufficient, but not necessary, for B A is neither necessary nor sufficient for B

Example 1 (~ A  ~ P ) ( A  P ) averaging (at least) fifty A is both necessary and sufficient for passing P A is necessary for P (~ A  ~ P ) and  A is sufficient for P ( A  P )

E is not sufficient for A Example 2 taking four exams E is necessary but not sufficient for getting an A A E is necessary for A (~E  ~A ) but  E is not sufficient for A ~ ( E  A )

Example 3 ( H  A )  getting a hundred H is sufficient but not necessary for getting an A A H is sufficient for A ( H  A ) but  E is not necessary for A ~ (~H  ~A )

A is not sufficient for P Example 4 attending class A is neither necessary nor sufficient for passing P A is not necessary for P ~ (~ A  ~ P ) and  A is not sufficient for P ~ ( A  P )

Complex Statements (Step by Step Procedure) 1. Identify the simple (atomic) statements; abbreviate them by upper case letters. 2. Identify all the connectives. Which are standard? Which are non-standard? 3. Rewrite the sentence replacing the simple statements by their abbreviations. Retain internal punctuation. 4. Identity the major connective. [Notice commas!] 5. If the major connective is standard, symbolize it; otherwise, paraphrase it and repeat 4. 6. Work on the constituent formulas resulting from 5 [go to 4]. 7. Substitute constituents back into overall formula. 8. Translate the formula back into English, and compare it with original sentence.

Example 1 if neither JAY nor KAY is working, then we will go on VACATION. 1. simple sentences J = Jay is working K = Kay is working V = we go on vacation 2. connectives if...then (standard) neither...nor (non-standard)

Example 1(contd.) ( neither J nor K )  V 3. first formula if neither J nor K, then V 4. main connective? 5. standard or non-standard? standard, so we symbolize it ( neither J nor K )  V

Example 1(contd.) ( not J and not K ) (~ J  ~ K ) 6. work on constituents ( neither J nor K )  V ( not J and not K ) (~ J  ~ K ) 7. substitute constituents (~ J  ~ K )  V 8. translate formula back into English if Jay is not working, and Kay is not working, then we will go on vacation.

Example 2 unless the exam is EASY, I will PASS only if I STUDY 1. simple sentences E = the exam is easy P = I pass S = I study 2. connectives unless (non-standard) only if (non-standard)

Example 2(contd.) 3. first formula unless E, P only if S 4. main connective 5. standard or non-standard non-standard, so we paraphrase it unless = if not if not E, then P only if S

Example 2(contd.) if not E, then P only if S 4. main connective 5. standard or non-standard standard, so we symbolize it ( not E )  ( P only if S ) 6. work on constituents ( not E )  ( P only if S ) ~ E ( not P if not S ) ( if not S, then not P ) (~ S  ~ P )

Example 2 (contd.) 7. Substitute constituents ~ E  (~ S  ~ P ) 8. Translate formula back into English if the exam is not easy, then I do not study, I will not pass

Example 3 if I am HAPPY only if I am DRUNK, then I am not HAPPY unless I am DRUNK. 1. simple sentences: H = I am happy D = I am drunk 2. connectives: if…then (standard) only if (non-standard) not (standard) unless (non-standard)

Example 3 (contd.) 3. first formula: if H only if D, then not-H unless D 4. main connective: if H only if D, then not-H unless D 5. standard or non-standard: standard, so we symbolize it ( H only if D )  ( not-H unless D )

Example 3 (contd.) 6. Work on constituents ( H only if D )  ( not-H unless D ) ( not H if not D ) (if not D, then not H ) (~ D  ~ H ) ( H only if D )  ( not-H unless D ) ( not-H if not D ) ( if not-D then not H )

Example 3 (contd.) 7. Substitute constituents (~ D  ~ H )  (~ D  ~ H ) 8. Translate formula back into English IF if I am not drunk then I am not happy THEN

Example 4 if I CONCENTRATE well only if I am ALERT, then provided I am WISE I will not FLY an airplane unless I am SOBER 1. simple sentences: C = I concentrate well A = I am alert W = I am wise F = I fly an airplane S = I am sober 2. connectives: if…then (standard) only if (non-standard) provided (non-standard) not (standard) unless (non-standard)

Example 4 (contd.) 3. first formula: if C only if A, then provided W, not F unless S 4. main connective: if C only if A, then provided W not F unless S 5. standard or non-standard standard, so we symbolize it ( C only if A )  ( provided W, not F unless S )

Example 4 (contd.) 6a. work on first constituent – antecedent ( C only if A )  ( provided W, not-F unless S ) ( not C if not A ) (if not A, then not C ) (~ A  ~ C )

Example 4 (contd.) 6b. work on second constituent – consequent ( C only if A )  (provided W, not-F unless S) main connective? provided W, not-F unless S paraphrase? if W, then ( not-F unless S ) W  ( not F unless S ) W  ( not F if not S ) W  ( if not S, then not F ) W  (~ S  ~ F )

Example 4 (contd.) substitute constituents (~ A  ~ C)  ( W  (~ S  ~ F) ) 8. Translate formula back into English IF if I am not alert then I do not concentrate well THEN if I am wise, then if I am not sober then I will not fly an airplane