Discrete tics Unit-I
-
Upload
vijayan-bharathi -
Category
Documents
-
view
280 -
download
2
Transcript of Discrete tics Unit-I
![Page 1: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/1.jpg)
Celestine Bernard
![Page 2: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/2.jpg)
LOGIC
Statement:
Any meaningful sentence.
Any sentence with subject, predicate
and verb
Ex:
1) Canada is a country
2)1+101=110
3) X=2 is a solution of =4.
![Page 3: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/3.jpg)
Compound proposition:
A proposition consists of two or more
simple propositions.
Ex:
Consider any two propositions.
1) Roses are red in color.
2) Shirt is blue in color
Then the compound proposition is
Roses are red and shirts are blue in
color.
![Page 4: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/4.jpg)
Connectors:
1. Disjunction:
Symbol: [V]
Operator name: And
Consider two propositions P and Q.
The symbolic representation will be
P V Q
Example:
P: It is raining today
Q: There are ten tables in this room.
P V Q: it is raining today or there are
Ten tables are in this room.
![Page 5: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/5.jpg)
Truth table:
P
Q P V Q
T T T
T F T
F T T
F F F
![Page 6: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/6.jpg)
2) Conjunction:
Symbol: [ʌ]
Operator name: OR
Explanation:
Consider any two propositions P and Q
Then the Symbolic representation will be
P Λ Q
Example:
P: It is raining today.
Q: there are ten tables in this room.
P Λ Q: It is raining today and there are ten
tables in this room.
![Page 7: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/7.jpg)
Q Λ P: there are ten tables in this room
And it is raining today.
Truth table:
P
Q P Λ Q
T T T
T F F
F T F
F F F
![Page 8: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/8.jpg)
3. Negation:
Symbol: ~
Operator name: negation
Explanation:
Consider a proposition P
The symbolic representation will be
~ P
Example:
P: London is a city
~P: London is not a city.
Truth table:
P ~P
T F
F T
![Page 9: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/9.jpg)
Conditional Propositions:
Symbol: P Q (If P then Q)
Truth table:
P
Q P → Q
T T T
T F F
F T T
F F T
Ex:
P: 3=8 …………………………….F
Q: 3+5=8 ….………………………T
P Q T
If 3=8 then 3+5=8.
![Page 10: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/10.jpg)
Biconditional propositions:
Symbol: P ↔ Q (P if and only if Q)
Truth table:
P
Q P ↔ Q
T T T
T F F
F T F
F F T
NOTE:
P → Q is direct proposition.
Q → P is transverse of P → Q.
~P → ~Q is inverse function.
~Q → ~P is contra positive.
![Page 11: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/11.jpg)
Q → P ≡ ~P → ~Q
(Transverse ≡ Inverse)
P → Q ≡ ~Q → ~P
(Direct Proposition ≡ Contra positive)
P ↔ Q ≡ (P → Q) ʌ (Q → P)
(Biconditional ≡Direct Prop., Λ transverse)
Examples:
1) Construct truth table:
Q Λ (P Q) P
P Q PQ QΛ(PQ) QΛ(PQ)P
T T T T T
T F F F T
F T T T F
F F F F T
![Page 12: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/12.jpg)
2) (P Q) Λ (Q P)
Truth table:
P Q PQ (QP) (PQ)Λ(QP)
T T T T T
T F T F F
F T F T F
F F T T T
3) ~( P Λ Q) ↔ (~P V ~Q)
P Q ~P ~Q (~PV~Q) PΛQ ~(PΛQ) ~(PΛQ)↔(~PV~Q)
T T F F F T F T
T F F T T F T T
F T T F T F T T
F F T T T F T T
![Page 13: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/13.jpg)
Logical Equivalence:
Suppose that compound propositions
P and Q are made up of the propositions
P1, P2,P3,…………………………………….… Pn
Then P ≡ Q provided that given any truth
values of
P1, P2, P3 ,……………………………………….P n
either P and Q are both true or P and Q
both false.
![Page 14: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/14.jpg)
Properties of Equivalence:
Idempotent law:
Disjunction Conjunction
P V P ≡ P P Λ P ≡ P
Associative law:
Disjunction
(P V Q) V R ≡ P V (Q V R) Conjunction
(P Λ Q) Λ R ≡ P Λ (Q Λ R)
Commutative law:
Disjunction
(P V Q) ≡ (Q V P) Conjunction (P Λ Q) ≡ (Q Λ P)
![Page 15: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/15.jpg)
Distributive law:
Disjunction
P V (Q Λ R) ≡ (P V Q) Λ (P V R) Conjunction P Λ (Q V R) ≡ (P Λ Q) V (P Λ R)
Identity law:
Disjunction Conjunction
T V P ≡ T T Λ P ≡ P F V P ≡ P F Λ P ≡ F
Component law:
Disjunction Conjunction
P V ~P ≡ T P Λ ~P ≡ F ~T ≡ F ~F ≡ T
![Page 16: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/16.jpg)
Absorption law:
Disjunction
P V( P V Q) P Conjunction P Λ( P Λ Q) P
DE Morgan’s law:
Disjunction
~(P V Q) ≡ ~P Λ ~Q Conjunction ~(P Λ Q) ≡ ~P V ~Q
Innovation law:
~(~P) ≡ P
![Page 17: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/17.jpg)
Tautologies and Contradictions:
Tautology:
A statement is called a tautology (or) a
universally valid formula (or) logically
truth when the last column of the
statement is true.
Cheat sheet value: Last column will be
true
Ex: Find the tautology and contradiction.
1) P (PVQ)
Sol:
P Q PVQ P(PVQ)
T T T T
T F T T
F T T T
F F F T
Ans: It’s a Tautology
![Page 18: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/18.jpg)
Contradiction:
A statement is called a contradiction if
the last column of the statement.
Cheat sheet value: Last column will be
false
Ex:
1) Find the tautology and contradiction.
(~Q Λ P) Λ Q
Sol:
P Q ~Q (~QΛP) (~QΛP)ΛQ
T T F T F
T F T T F
F T F T F
F F T F F
Ans: It’s a contradiction
![Page 19: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/19.jpg)
Tautology (or) Contradiction:
Cheat sheet value: Last column will have
Both (True and False) values
Ex:
1) Find the tautology and contradiction
P Q PΛQ (PΛQ)↔P
T T T T
T F F F
F T F T
F F F T
Ans:
It’s neither tautology nor contradiction.
![Page 20: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/20.jpg)
Exercise:
1) Construct truth table for the
following. And find whether it is
tautology or contradiction.
(P~P)~P
(~PQ)(QP)
[P(QR)][(PQ)(PR)]
[PΛ(PQ)](PΛQ)
[(P↔Q)]↔*(PΛQ)V(~PΛ~Q)+
![Page 21: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/21.jpg)
Substitution instance:
A formula X is called a substitution
instance of another formula Y, If X can be
obtain from Y by substituting formula’s for
some variables of Y.
Ex:
P V Q ≡ ~(~P) V ~(~Q)
Important substitution formulae:
P ↔ Q ≡ (P→Q) V (Q P)
P Q ≡ ~P V Q
Cheat sheet value:
Any substitution instance of a
tautology is tautology
![Page 22: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/22.jpg)
Replacement process:
It is just like substitution instance.
You can replace a formula by another
formula if it equals.
Ex:
P Λ Q (P V Q) Λ Q
![Page 23: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/23.jpg)
Example sum:
Show that
P(QR) (P(~Q VR)) ((PΛQ) R)
Sol:
There are three parts. Consider any two
statements and prove.
P (QR) P (~Q V R)
~P V (~QVR)
Using PQ ~P V Q
(~P V~Q) V R
Using associative property
(P V Q) V R ≡ P V (Q V R)
~ (P Λ Q) V R
Using DE Morgan’s law
~P V ~Q ~ (P Λ Q)
![Page 24: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/24.jpg)
(P Λ Q) R
Using ~P V Q P Q
2. Show that
{(PVQ) Λ~ [PΛ (~QV~R)]} V
{(~PΛ~Q) V (~PΛ~R)}
is a tautology
Sol:
This statement has two parts. So take
the first part and reduce it. Same way do it
for the second one. Finally combine those
two equations and make that as true.
(PVQ)Λ [~PΛ (~Q V~R)]
(PVQ) Λ ~ (~P Λ ~ (Q Λ R))
Using DE Morgan’s law
(PVQ) Λ ~~ (PV (QΛR))
![Page 25: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/25.jpg)
Using DE Morgan’s law
(PVQ)Λ (PV (QVR))
Using ~~PP
(PVQ)Λ * (PVQ) Λ (PVR) +
Using distributive law
*(PVQ) Λ (PVQ)+ Λ (PVR)
Using associative law
(PVQ)Λ (PVR) ……………………….(1)
Now,
Take the second part and reduce it.
*(~PΛ~Q) V (~PΛ~R)+
~ (PVQ) V ~ (PVR)
Using DE Morgan’s law
~ *(PVQ) Λ (PVR)+…………………. (2)
![Page 26: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/26.jpg)
Combine equations 1 & 2
[(PVQ) Λ (PVR)]V ~ *(PVQ) Λ (PVR)+
We know that
P V ~P ≡ T
PV~P
T
So it is a tautology.
For more formulae………….
Refer CHEAT SHEET
Attached with this short notes
![Page 27: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/27.jpg)
Duality law:
Two formulae A and A* are said to be
duals of each other if either one can be
obtained from other one by replacing
V by Λ and Λ by V.
The connectives V and Λ are known as
duals of each other.
Cheat sheet value:
Replace...
V by Λ
Λ by V
T by F
F by T
![Page 28: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/28.jpg)
Example:
Write the duals for the following.
1) (PVQ) Λ R (PΛQ) V R
2) (PΛQ) V T (PVQ) Λ F
Sol:
1) (PΛQ)V R (PVQ) Λ R
2) (PVQ)Λ F (PΛQ) V T
![Page 29: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/29.jpg)
Cheat sheet value:
Proving methods:
Consider a question like this
Prove that A B CQ
Method I
By using property of equivalence
Ex: Associative law, DE Morgan’s law…
Method II
By using tautology
(i.e) make AB and CD as true.
Ex: AB T
CD T
Here you can write ABCD
![Page 30: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/30.jpg)
Example for Method 2:
Prove that
P (QP) [~P (PQ)]
Sol:
Method II
P (QP) ~P V (~QVP)
(~PVP) V ~Q
T V~Q
T ………………………..(1)
[~P(PQ)]~P(~PVQ)
PV(~PVQ)
(P V~P) VQ
T VQ
T …………………………(2)
From 1 & 2, P (QP) [~P (PQ)]
![Page 31: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/31.jpg)
Example for method I:
Prove that
P(QVR) (P Q) V(PR)
Sol:
R.H.S (PQ)V(PR)
(~PVQ) V (~PVR)
(~PV~P) V(QVR)
~P V (QVR)
P(QVR)
L.H.S
![Page 32: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/32.jpg)
Tautological implications:
A statement M is said to be
tautologically implied to a statement N if
and only if MN is a tautology.
Tautological implications:
P Λ Q P
P Λ Q Q
P P V Q
~P P Q
Q P Q
~ (P Q) P
~ (P Q) ~Q
P Λ ( P Q) Q
~ Q Λ ( P Q) ~P
~P Λ ( P V Q) Q
( P Q) Λ (Q R) PR
( P V Q) Λ (P R) Λ (Q R) R
![Page 33: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/33.jpg)
Cheat sheet value:
PΛQ P
Antecedent consequent
Proof:
1) PΛQ P
Sol:
P Q P V Q P V QP
T T T T
T F F T
F T F T
F F F T
It is a tautology.
Therefore PΛQP
Same way we can prove all statements.
![Page 34: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/34.jpg)
Normal forms:
DNF :( Disjunctive Normal Form)
Cheat sheet value:
DNF is not unique
It should have only Λ and V symbols
It shouldn’t have any other symbols
Ex: and ↔
Syntax:
(… A ...) V (...B...)
Example:
Find DNF of the following
(I) PΛ(PQ)
(II) P[(PQ)Λ~(~QV~P)+
![Page 35: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/35.jpg)
Sol:
(i) PΛ(PQ)
PΛ (~PVQ)
(PV~P) V (PVQ)……………….. (DNF)
(ii) P[(PQ)Λ(~QΛ~P)+
Sol:
~PV [(PQ) Λ (~QΛ~P)+
~PV *(~PVQ) Λ~~ (QVP)+
~PV *(~PVQ) Λ (PVQ)+
~PV *(~PVQ) Λ (PΛQ)+
~PV ,*~PΛ (PΛQ)+ V *QΛ (PΛQ)+-
Using distributive law
~PV *(~PΛPΛQ) V (QΛPΛQ)+
~PV *(TΛQ) V (PΛQ)+
~PV *F V (QΛP)+
~PV (QΛP) …………………… (DNF)
![Page 36: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/36.jpg)
Conjunctive normal form (CNF):
Cheat sheet value:
(… A …) Λ (… B …)
It should have only Λ and V symbols
It shouldn’t have any other symbols
Ex: and ↔
Example:
Find the CNF for the following
a) PΛ(PQ)
b) *PΛ~(PVR)+V , *(PΛQ)V~R+ Λ P-
Sol:
a) PΛ(PQ)
PΛ (~P V Q)………………………. (CNF)
![Page 37: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/37.jpg)
b) *PΛ~(PVR)+V { *(PΛQ)V~R+ Λ P-
*PΛ~ (PVR)+V ,*(PΛQ) V~R+ Λ P-
*PΛ (~PΛ~R)+V ,*(PΛQ) ΛP] V
[~RΛP+- …using distributive law
*(PΛ~P) Λ~R+V ,*(PΛP) ΛQ+ V
*~RΛP+-
*FΛ~R+ V *(PΛQ) V (~RΛP)+
F V *(PΛQ) V (~RΛP)+
(PΛQ) V (~RΛP) ……………….. (CNF)
![Page 38: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/38.jpg)
Principle disjunctive normal form (PDNF):
Let P and Q be two statement
variables. Let us construct all possible
formulas which consist of P, ~P, Q, ~Q.
The possible forms are
PΛQ, ~PΛQ, PΛ~Q, ~PΛ~Q
These formulas are called minterms or
Boolean conjunctions.
For a given formula, an equivalent
formula consisting of disjunctions of
minterms only is known as its PDNF.
Such a normal form is also called the
Sum of products canonical form.
Cheat sheet value(PDNF):
Syntax:
(…V…) Λ (…V…) Λ(…V…)
![Page 39: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/39.jpg)
Obtain PDNF for the following.
1) ~P V Q
2) (PΛQ) V (~PΛR) V (QΛR)
Sol:
1) ~P V Q
Introduce a variable T
(~PVT) V (QVT)
Introduce the missing variable using
component law.
[~PV (~QVQ)] V [QV (PV~P)]
(~PΛQ) V (~PΛ~Q) V (QΛP)V(QΛ~P)
Using distributive law
(~PΛQ) V (~PV ~Q) V (QΛP)
If a minterm repeats just negate 1
PDNF
![Page 40: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/40.jpg)
2) (PΛQ) V (~PΛR) V (QΛR)
Introduce the new variable T
*(PΛQ)ΛT+V*(~PΛR)ΛT+ V*(QΛR) ΛT+
*(PΛQ) Λ(RV~R)+V*(~PΛR)Λ(QV~Q)+
V *(QΛR)(PV~P)+
(PΛQΛR) V (PΛQΛ~R) V(~PΛRΛ~Q)
V (~PΛRΛ~Q) V(QΛRΛP)V (QΛRΛ~P)
Using distributive law.
If a minterm repeats just negate 1.
(PΛQΛR) V (PΛQΛ~R) V
(~PΛQΛR) V (~PΛ~QΛR)
PDNF
![Page 41: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/41.jpg)
Principle conjunctive normal form (PCNF):
Let us consider two variables P & Q
Let us write all possible maxterms.
The maxterms are,
P V Q
~P V ~Q
~P V Q
P V ~Q
For a given formula, an equivalent formula
consisting of conjunctions of maxterms
only is known as its PCNF. This form is also
called as the product of sums of canonical
form.
Cheat sheet value:
Syntax:
(…Λ…) V (…Λ…) V(…Λ…)
![Page 42: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/42.jpg)
Example:
Obtain PCNF for the following.
P↔Q
Sol:
Step 1: construct the truth table
P Q Minterms P↔Q
T T P Λ Q T T F P Λ~Q F F T ~PΛQ F F F ~PΛ~Q T
*T=P & F=~P ⌂ *T=Q & F=~Q
Step 2:
Write down the minterms which has true
value and make that as A
A (PΛQ) V (~PΛ~Q)
![Page 43: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/43.jpg)
Step 3:
Find ~A
~A (~PΛ ~Q) V (P ΛQ)
*P=~P *Q=~Q
Step 4:
Find ~ (~A)
~(~A)~{ (~PΛ ~Q) V (P ΛQ)-….PCNF of A
For more theory refer discrete maths I-2
![Page 44: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/44.jpg)
Celestine Bernard
Discrete mathematics
![Page 45: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/45.jpg)
Cheat Sheet
Disjunction:
Truth table:
P
Q P V Q
T T T
T F T
F T T
F F F
Conjunction:
Truth table:
P
Q P Λ Q
T T T
T F F
F T F
F F F
![Page 46: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/46.jpg)
Negation:
Truth table:
P ~P
T F
F T
Conditional propositions:
Truth table:
P
Q P → Q
T T T
T F F
F T T
F F T
![Page 47: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/47.jpg)
Biconditional propositions:
Truth table:
P
Q P ↔ Q
T T T
T F F
F T F
F F T
NOTE:
P → Q is direct proposition.
Q → P is transverse of P → Q.
~P → ~Q is inverse function.
~Q → ~P is contra positive.
![Page 48: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/48.jpg)
Q → P ≡ ~P → ~Q
(Transverse ≡ Inverse)
P → Q ≡ ~Q → ~P
(Direct Proposition ≡ Contra positive)
P ↔ Q ≡ (P → Q) ʌ (Q → P)
(Biconditional ≡Direct Prop., Λ transverse)
![Page 49: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/49.jpg)
Properties of Equivalence:
Idempotent law:
Disjunction Conjunction
P V P ≡ P P Λ P ≡ P
Associative law:
Disjunction
(P V Q) V R ≡ P V (Q V R) Conjunction (P Λ Q) Λ R ≡ P Λ (Q Λ R)
Commutative law:
Disjunction
(P V Q) ≡ (Q V P) Conjunction (P Λ Q) ≡ (Q Λ P)
![Page 50: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/50.jpg)
Distributive law:
Disjunction
P V (Q Λ R) ≡ (P V Q) Λ (P V R) Conjunction P Λ (Q V R) ≡ (P Λ Q) V (P Λ R)
Identity law:
Disjunction Conjunction
T V P ≡ T T Λ P ≡ P F V P ≡ P F Λ P ≡ F
Component law:
Disjunction Conjunction
P V ~P ≡ T P Λ ~P ≡ F ~T ≡ F ~F ≡ T
![Page 51: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/51.jpg)
Absorption law:
Disjunction
P V( P V Q) P Conjunction P Λ( P Λ Q) P
DE Morgan’s law:
Disjunction
~(P V Q) ≡ ~P Λ ~Q Conjunction ~(P Λ Q) ≡ ~P V ~Q
Innovation law:
~(~P) ≡ P
![Page 52: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/52.jpg)
Substitution instance:
P ↔ Q ≡ (P→Q) V (Q P)
P Q ≡ ~P V Q
Tautological implications:
P Λ Q P
P Λ Q Q
P P V Q
~P P Q
Q P Q
~ (P Q) P
~ (P Q) ~Q
P Λ ( P Q) Q
~ Q Λ ( P Q) ~P
~P Λ ( P V Q) Q
( P Q) Λ (Q R) PR
( P V Q) Λ (P R) Λ (Q R) R
![Page 53: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/53.jpg)
Implication, Equivalence formula’s for
Inference theory:
Implication formulae:
P P V Q ……………………………….(I -1)
Q P V Q ……………………………….(I-2)
P Λ Q P ………………………………..(I-3)
P Λ Q Q ……………………………….(I-4)
P , PQ Q …………………………..(I-5)
~Q , PQ ~P ……………………….(I-6)
~P, P V Q Q ………………………….(I-7)
P Q, Q R PR ………………(I-8)
P , Q P Λ Q ……………………………(I-9)
Q PQ ……………………………….(I-10)
P V Q , PR , QR R ………….(I-11)
~P PQ ……………………………….(I-12)
~(P Q) P …………………………..(I-13)
~(P Q) ~P ………………………..(I-14)
![Page 54: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/54.jpg)
Equivalence formulae:
~ ~ P P …………………………………(E-1)
P Q ~P V Q ……………………….(E-2)
P Q ~Q ~P ……………………(E-3)
P ↔ Q (P Q) Λ (QP) …….(E-4)
P (Q R) (P Λ Q) R …….(E-5)
~(P Λ Q) (~P V ~Q) ………………..(E-6)
(P ↔ Q ) (P Λ Q) V (~P V ~Q)…(E-7)
Quantifiers:
There are two types of quantifiers.
Universal quantifier
Symbol: (Ɏ) Meaning: All
Existential quantifier
Symbol :( ƭ) Meaning: some
![Page 55: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/55.jpg)
Equivalent formulae :( Quantifiers)
(Ɏx) *P(x)+ ~(Ɏx) *~P(x)+
(Ɏx) *~P(x)+ ~(ƭx) P(x)
~(Ɏx) *P(x)+ (ƭx)*~P(x)+
~(Ɏx) *~P(x)+ (ƭx) *~P(x)+
(ƭx) *A(x) V B(x)+ *(ƭx)A(x) V (ƭx)B(x)+
(Ɏx)*A(x) Λ B(x)+ *(Ɏx)A(x)V(Ɏx)B(x)+
(Ɏx)*A Λ B(x)+ A V (Ɏx)B(x)
(ƭx)*A Λ B(x)+ A Λ (ƭx) B(x)
(Ɏx) A(x) B (ƭx) (A-1(x)B)
(ƭx) A(x) B (Ɏx)*A(x)B]
A(Ɏx)B(x) (Ɏx)*AB(x)]
A(ƭx)B(x) (ƭx)* AB(x)]
Implication formulae:
(Ɏx)A(x) V(Ɏx)B(x) (Ɏx)*A(x) VB(x)+
(ƭx) *A(x) Λ B(x)+ (ƭx) A(x) Λ (ƭx) B(x)
P(y) (ƭx)P(x)
(Ɏx)P(x) (ƭx) P(x)
(Ɏx)*P(x)Q(x)] (Ɏx)P(x)(Ɏx)Q(x)
![Page 56: Discrete tics Unit-I](https://reader036.fdocuments.us/reader036/viewer/2022081716/54fe1c3f4a7959f4248b4c2a/html5/thumbnails/56.jpg)
Theory of inference:
Rules:
1. Universal specification(UF):
(Ɏx) A(x) A(y)
2. Universal generation(UG):
A(y) (Ɏx)A(x)
3. Existential specification(ES):
(ƭx)A(x) A(y)
4. Existential generation(EG):
A(y) (ƭx)A(x)