DMS UNIT-1 ppt.pptx

1. Discrete Structures
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2. Compound Propositions
Producing new propositions from existing propositions.
Logical Operators or Connectives
1. Not 
2. And ˄
3. Or ˅
4. Exclusive or 
5. Implication 
6. Biconditional 
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3. Compound Propositions
Negation of a proposition
Let p be a proposition. The negation of p, denoted by
 p (also denoted by ~p), is the statement
“It is not the case that p”.
The proposition  p is read as “not p”. The truth
values of the negation of p,  p, is the opposite of the
truth value of p.
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4. Examples
1. Find the negation of the following proposition
p : Today is Friday.
The negation is
 p : It is not the case that today is Friday.
This negation can be more simply expressed by
 p : Today is not Friday.
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5. Examples
2. Write the negation of
“6 is negative”.
The negation is
“It is not the case that 6 is negative”.
or “6 is nonnegative”.
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6. Truth Table (NOT)
• Unary Operator, Symbol: 
p p
true false
false true
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7. Conjunction (AND)
Definition
Let p and q be propositions. The conjunction
of p and q, denoted by p˄q, is the proposition
“p and q”.
The conjunction p˄q is true when p and q are
both true and is false otherwise.
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8. Examples
1. Find the conjunction of the propositions p and q, where
p : Today is Friday.
q : It is raining today.
The conjunction is
p˄q: Today is Friday and it is raining today.
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9. Truth Table (AND)
• Binary Operator, Symbol: 
p q pq
true true true
true false false
false true false
false false false
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10. Disjunction (OR)
Definition
Let p and q be propositions. The disjunction
of p and q, denoted by p˅q, is the proposition
“p or q”.
The disjunction p˅qis false when both p and
q are false and is true otherwise.
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11. Examples
1. Find the disjunction of the propositions p and q,
where
p : Today is Friday.
q : It is raining today.
The disjunction is
p˅q: Today is Friday or it is raining today.
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12. Truth Table (OR)
• Binary Operator, Symbol: 
p q p  q
true true true
true false true
false true true
false false false
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13. Exclusive OR (XOR)
Definition
Let p and q be propositions. The exclusive or
of p and q, denoted by pq, is the proposition
“pq”.
The exclusive or, p  q, is true when exactly
one of p and q is true and is false otherwise.
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14. Examples
1. Find the exclusive or of the propositions p and q,
where
p : Atif will pass the course CSC102.
q : Atif will fail the course CSC102.
The exclusive or is
pq : Atif will pass or fail the course CSC102.
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15. Truth Table (XOR)
• Binary Operator, Symbol: 
p q pq
true true false
true false true
false true true
false false false
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16. Examples (OR vs XOR)
The following proposition uses the (English) connective
“or”. Determine from the context whether “or” is intended
to be used in the inclusive or exclusive sense.
1. “Nabeel has one or two brothers”.
A person cannot have both one and two brothers.
Therefore, “or” is used in the exclusive sense.
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17. Examples (OR vs XOR)
2. To register for BSC you must have passed
the qualifying exam or be listed as an Math
major.
Presumably, if you have passed the qualifying exam and
are also listed as an Math major, you can still register for
BCS. Therefore, “or” is inclusive.
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18. Composite Statements
Statements and operators can be combined in any
way to form new statements.
p q p q (p)(q)
true true false false false
true false false true true
false true true false true
false false true true true
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19. Translating English to Logic
I did not buy a lottery ticket this week or I bought a
lottery ticket and won the million dollar on Friday.
Let p and q be two propositions
p: I bought a lottery ticket this week.
q: I won the million dollar on Friday.
In logic form
p(pq)
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20. 20

21. Conditional Statements
Implication
Definition: Let p and q be propositions. The conditional
statement p  q, is the proposition “If p, then
q”.
The conditional statement p  q is false when
p is true and q is false and is true otherwise.
where p is called hypothesis, antecedent or premise.
q is called conclusion or consequence
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22. Implication (if - then)
• Binary Operator, Symbol: 
P Q P  Q
true true true
true false false
false true true
false false true
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23. Conditional Statements
Definition: Let p and q be
biconditional statement pq, is
propositions. The
the
proposition “p if and only if q”.
The biconditional (bi-implication) statement p
 q is true when p and q have same truth
values and is false otherwise.
Biconditional Statements
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24. Biconditional (if and only if)
• Binary Operator, Symbol: 
P Q P  Q
true true true
true false false
false true false
false false true
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25. Composite Statements
• Statements and operators can be combined in any way to
form new statements.
P Q P Q (P)(Q)
true true false false false
true false false true true
false true true false true
false false true true true
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26. Equivalent Statements
P Q (PQ) (P)(Q) (PQ)(P)(Q)
true true false false true
true false true true true
false true true true true
false false true true true
•Two statements are called logically equivalent if and only if (iff) they
have identical truth tables
• The statements (PQ) and (P)(Q) are logically equivalent,
because (PQ)(P)(Q) is always true.
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27. Tautologies and Contradictions
• Tautology is a statement that is always true regardless of the truth
values of the individual logical variables
• Examples:
• R(R)
• (PQ)  (P)(Q)
• If S  T is a tautology, we write S  T
.
• If S  T is a tautology, we write S  T
.
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28. Tautologies and Contradictions
• A Contradiction is a statement that is always false regardlessof
the truth values of the individual logical variables
Examples
• R(R)
• ((PQ)(P)(Q))
• The negation of any tautology is a contradiction, and
the negation of any contradiction is a tautology.
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29. Exercises
•We already know the following tautology:
•(PQ)  (P)(Q)
•Nice home exercise:
•Show that (PQ)  (P)(Q).
•These two tautologies are known as De Morgan’s laws.
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30. Logical Equivalence
Definition
Two proposition form are called logically equivalent if
and only if they have identical truth values for each
possible substitution of propositions for their
proposition variable.
The logical equivalence of proposition forms P
and Q is written
P ≡ Q
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31. 1. Construct the truth table for P.
2. Construct the truth table for Q using the
same proposition variables for identical
component propositions.
3. Check each combination of truth values of
the proposition variables to see whether the
truth value of P is the same as the truth
value of Q.
Equivalence of Two Compound
Propositions P and Q
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32. a. If in each row the truth value of P is the
same as the truth value of Q, then P and Q
are logically equivalent.
b. If in some row P has a different truth value
from Q, then P and Q are not logically
equivalent.
Equivalence Check
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33. Example
• Prove that ¬ (¬p)≡ p
Solution
As you can see the corresponding truth values of p
and ¬ (¬p) are same, hence equivalence is justified.
p ¬p ¬ (¬p)
T F T
F T F
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34. Example
Show that the proposition forms ¬(pq) and ¬p  ¬q
are NOT logically equivalent.
Here the corresponding truth values
differ and hence equivalence does
not hold
p q ¬p ¬q (pq) ¬(pq) ¬p¬q
T T F F T F F
T F F T F T F
F T T F F T F
F F T T F T T
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35. De Morgan’s laws
De Morgan’s laws state that:
The negation of an and proposition is
logically equivalent to the or proposition in
which each component is negated.
The negation of an or proposition is logically
equivalent to the and proposition in which
each component is negated.
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36. Symbolically (De Morgan’s Laws)
1. ¬(pq) ≡ ¬p¬q
2. ¬(pq) ≡ ¬p¬q
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37. Applying De-Morgan’s Law
Question: Negate the following compound Propositions
1. John is six feet tall and he weights at least 200
pounds.
2. The bus was late or Tom’s watch was slow.
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38. Solution
a) John is not six feet tall or he weighs less
than 200 pounds.
b) The bus was not late and Tom’s watch was
not slow.
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39. Inequalities and De Morgan’s Laws
Question Use De Morgan’s laws to write the negation of
-1< x  4
Solution: The given proposition is equivalent to
-1 < x and x  4,
By De Morgan’s laws, the negation is
-1 ≥ x or x > 4.
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40. Tautology and Contradiction
Definition A tautology is a proposition form that is
always true regardless of the truth values of the
individual propositions substituted for its proposition
variables. A proposition whose form is a tautology is
called a tautological proposition.
Definition A contradiction is a proposition form that is
always false regardless of the truth values of the
individual propositions substituted for its proposition
variables. A proposition whose form is a contradiction is
called a contradictory proposition.
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41. Example
Show that the proposition form p¬p is a
tautology and the proposition form p¬p is a
contradiction.
Exercise: If t is a tautology and c is
contradiction, show that pt≡p and pc≡c?
p ¬p p ¬p p ¬p
T F T F
F T T F
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42. 1. Commutative laws
pq ≡ qp ; pq ≡ qp
2. Associative laws
p  (q  r) ≡ (p q)  r ; p(q r) ≡ (pq)r
3. Distributive laws
p  (q r ) ≡ (p  q)  (p  r)
p  (q  r) ≡ (p  q)  (p  r)
Laws of Logic
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43. Laws of Logic
4. Identity laws
p  t ≡ p ; pc ≡ p
5. Negation laws
p¬p ≡ t ; p  ¬p ≡ c
6. Double negation law
¬(¬p) ≡ p
7. Idempotent laws
p  p ≡ p ; pp ≡ p
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44. Laws of Logic
8. Universal bound laws
p  t ≡ t ; p  c ≡ c
9. Absorption laws
p (pq) ≡ p ; p (p  q) ≡ p
10. Negation of t and c
¬t ≡ c ; ¬c ≡ t
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45. Exercise
Using laws of logic, show that
⌐(⌐p  q) (p  q) ≡ p.
Solution
Take ⌐(⌐p  q) (p  q)
≡ (⌐(⌐p)  ⌐q) (p  q), (by De Morgan’s laws)
≡ (p  ⌐q) (p  q), (by double negative law)
≡ p (⌐q  q), (by distributive law)
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46. contd…
≡ p (q  ⌐q), (by the commutative law)
≡ p  c, (by the negation law)
≡ p, (by the identity law)
Skill in simplifying proposition forms is useful in
constructing logically efficient computer programs
and in designing digital circuits.
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47. Another Example
Prove that ¬[r ∨(q ∧(¬r →¬p))] ≡ ¬r ∧(p∨¬q)
¬[r ∨(q ∧(¬r → ¬p))]
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≡ ¬r ∧¬(q ∧(¬r → ¬p)),
≡ ¬r ∧¬(q ∧(¬¬r ∨¬p)),
≡ ¬r ∧¬(q ∧(r ∨¬p)),
≡ ¬r ∧(¬q ∨¬(r ∨¬p)),
≡ ¬r ∧(¬q ∨(¬r ∧p)),
≡ (¬r ∧¬q) ∨(¬r ∧(¬r ∧p)),
≡ (¬r ∧¬q) ∨((¬r ∧¬r) ∧p),
≡ (¬r ∧¬q)∨(¬r ∧p),
≡ ¬r ∧(¬q ∨p),
≡ ¬r ∧(p ∨¬q),
De Morgan’s law
Conditional rewritten as disjunction
Double negation law
De Morgan’s law
De Morgan’s law, double negation
Distributive law
Associative law
Idempotent law
Distributive law
Commutative law

48. Lecture Summery
• Logical Connectives
• Truth Tables
• Compound propositions
• Translating English to logic and logic to English.
• Logical Equivalence
• Equivalence Check
• Tautologies and Contradictions
• Laws of Logic
• Simplification ofD
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