Assignement of discrete mathematics
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Discrete mathematics concerns processes that consist of a sequence of individual steps. Logic is the study of principles and methods that distinguish between valid and invalid arguments. A statement is a declarative sentence that is either true or false. Simple statements can be combined using logical connectives like AND, OR, and NOT to form compound statements. Truth tables specify the truth value of compound propositions based on the truth values of their constituent propositions and can be used to analyze compound statements. De Morgan's laws describe the relationships between logical connectives when negating compound propositions.
Assignement of discrete mathematics
- 1. Discrete Mathematics Defecation of Discrete Mathematics: Concerns processesthatconsist of a sequence of individual step are calledDiscrete Mathematics Continuous Discrete LOGIC: Logic studyof principlesandmethodthatdistinguishes/differenceb/w validandinvalidargumentis calledlogic. Simple statement: A statementisdeclarativesentence thatiseithertrue orfalse butnotboth. A statementisalsoreferredtoasa proposition Examples: 1. If a propositionistrue thenwe saythata value istrue. 2. Andif propositionisfalse thenwe saythattruth value isfalse. 3. Truth & false are denotedbyT and F Examples: Propositions: NotProposition: 1. Grass is green. Close the door. 2. 4+3=6. X isgreaterthan 2. 3. 4+2=6. He isveryrich. Rules: If the sentence isprecededbyothersentencesthatmake the pronounorvariable referenceclear,then the sentence isa statement.
- 2. Examples: False True I. X=1 Bill gatesisan American II. x>3 He is veryRich III. x>3 is a statementwithtruth-value He is veryrichis a statementwith truth-value UNDERSTANDINGSTATEMENTS: 1. X+2 ispositive Nota statement 2. May I come in? Nota statement 3. Logic isinteresting A statement 4. It ishot today A statement Compound Statement: Simple statementcouldbe usedtobuildacompoundstatement. Examples: 1. 3+2=5 and Multan isa city of Pakistan 2. The grass is greenor itis hottoday 3. Ali isnot veryrich And,Not,ORare called Logical Connectives. SYMBOLICREPRESENTATION: Statementis symbolicallyrepresentedbyletterssuchas“p,q, r… EXAMPLES: p=”Multan isa cityof Pakistan” q=”17 is divisible by3” CONNECTIV MEANING SYMBOL CALLED NEGATION NOT ˜ TILDE CONJUNCTION AND ^ HAT DISJUNCTION OR v VEL CONDITIONAL IF……THEN ARROW BICONDITIONAL IF ANDONLY IF DOUBLE ARROW
- 3. EXAMPLES: p=”Multan isa cityof Pakistan” q=”Ali isa Muslim” p ^ q=”Multanis a city of Pakistan”AND“Ali isa Muslim” p v q=” Multan isa cityof Pakistan”OR”Ali isa Muslim” ˜p=”Multanis nota cityof Pakistan” TRANSLATINGFROMENGLISHTO SYMBOLIS: Let p=”itis cold”,AND“it isAli” SENTENCE SYMBOLIC 1. It isnot cold ˜P 2. It iscold ANDAli P ^ q 3. It iscold OR Ali p v q 4. It isNOT coldBUT Ali ˜p ^ q COMPOUNDSTATEMENTEXAMPLES: Let a=”Ali isHealthy” b=”Ali isWealthy” c=”Ali is Wise” i. Ali ishealthyANDwealthyButNOTwise. (a ^ b) ^ (˜c) ii. Ali isNOT healthyBUThe iswealthyANDwise. (˜a) ^ (b^ c) iii. Ali isNEITHER healthy,WealthyNORwise. (˜a ^ ~ b v ~ c) TRANSLATINGFROMSYMBOLSTO ENGLISH: Let: m=”Ali isa good inmath” c=”Ali is a com. science student” I. ~ C Ali is“NOT” com. Science student. II. C v m Ali iscom. Science student”OR“goodinmath. III. M ^ ~ c Ali isgoodin math” BUT AND NOT“a com. Science student.
- 4. WHAT IS TRUTH TABLE? A truthtable specifiesthe truthvalue of acompoundpropositionforall possibletruthvaluesof its constituentproposition. A convenientmethodforanalyzingacompoundstatement istomake a truth table toit NEGATION (~) If p=statementvariable,thennegationof p“NOTp”, isdenotedby“~p” If p istrue,~p is false If p isfalse ~p istrue TRUTH TABLE FOR ~P P ~P T F F T CONJUNCTION (^) If p andq is statementthenconjunctionis“pand q” Denotedby“p ^q” If p andq are true thentrue If both or eitherfalse thenFalse P ^q P q P ^q T T T T F F F T F F F F DISJUNCTION (v) If P and q is statementthen“por q” Denotedby“p v q” If both are false thenfalse If both or eitheristrue thentrue
- 5. P v q P q P v q T T T T F T F T T F F F Truth Table for this statement ~p^ q P q ~p ~p ^q T T F F T F F F F T T T F F T F Truth Table for ~p^ (qv ~r) P q r ~r (q v ~r) ~p ~p ^(q v ~r) T T T F T F F T T F T T F F T F T F F F F T F F T T F F F T T F T T T F T F T T T T F F T F F T F F F F T T T T Truth table for (p v q) ^ ~ (p^q) P q (p v q) (p ^ q) ~(p ^q) (p v q)^~(p ^q) T T T T F F T F T F T T F T T F T T F F F F T F Double negationproperty ~ (~p) =p P (~p) ~(~p) T F T F T F So itis clearthat “p” and double negationof “p”isequal. Example Englishtosymbolic P= I am Umair Shah
- 6. ~p= I am not Umair Shah ~ (~p) = I am Umair Shah So itis clearthat double negationof “p”isalsoequal to “p”. ~ (p^q) & ~p ^~q are not Equal. P q (p ^q) ~(p ^q) ~p ~q ~p ^~q T T T F F F F T F F T F T F F T F T T F F F F F T T T T So itis clearthat “~ (p^q) & ~p ^~q” are not equal De Morgan’sLaw 1. The negationof “AND” statementislogicallyequivalenttothe “OR” statementinwhicheach componentisnegated. Symbolically~(p ^q) = ~p v ~q P q P ^q ~(p ^q) ~p ~q ~p v ~q T T T F F F F T F F T F T T F T F T T F T F F F T T T T So itis clearthat ~ (p^q) = ~p v ~q are logicallyequivalent 2. The negationof “OR” statementislogicallyequivalenttothe “AND”statementinwhichcomponent isnegated. Symbolically~(p v q) = ~p ^ ~q P q (P v q) ~(p v q) ~p ~q ~p ^ ~q T T T F F F F T F T F F T F F T T F T F F F F F T T T T So itis clearthat ~ (pv q) = ~p ^ ~q is Equal. Application Negationforeachof the following: a. The fanis slowor itis veryhot
- 7. b. Ali isfitor Awaisis injured. Solution: a. The fan isnot slow“AND”it is not veryhot b. Ali is not fit“AND” Awaisisnotinjured InequalitiesandDE MORGANESlaw: Exercise: 1. (p ^q) ^ r = P^(q ^ r) p q r (p ^q) (p ^q)^r (q ^r) P^(q ^r) T T T T T T T T T F T F F F T F T F F F F T F F F F F F F T T F F T F F T F F F F F F F T F F F F F F F F F F F So itclearsthat Colum5 and Colum7 are equal 2. (P ^q) v r = p ^ (qv r)????? P q r (p ^q) (p ^q) v r (q v r) P ^(q v r) T T T T T T T T T F T T T T T F T F T T T T F F F F F F F T T F T T F F T F F F T F F F T F T T F F F F F F F F So itclear that Colum5 and Colum7 are not equal.