Syed Umair

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.