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 negation, conjunction, and disjunction 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.

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.