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Logical Equivalence. Commutative laws: p  q = q  p, p  q = q  p Associative laws: (p  q)  r = p  (q  r), (p  q)  r = p  (q  r) Distributive laws: p  (q  r) = (p  q)  (p  r) p  (q  r) = (p  q)  (p  r)

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Logical equivalence
Logical Equivalence

  • Commutative laws: p  q = q  p, p  q = q  p

  • Associative laws: (p  q)  r = p  (q  r), (p  q)  r = p  (q  r)

  • Distributive laws: p  (q  r) = (p  q)  (p  r)

    p  (q  r) = (p  q)  (p  r)

  • Identity laws: p  t = p, p  c = p

  • Negation laws: p  ~p = t, p  ~p = c

  • Double negative law: ~(~p) = p

  • Idempotent laws: p  p = p, p  p = p

  • De Morgan’s laws: ~(p  q) = ~p  ~q, ~(p  q) = ~p  ~q

  • Universal bound laws: p  t = t, p  c = c

  • Absorption laws: p  (p  q) = p, p  (p  q) = p

  • Negation of t and c: ~t = c, ~c = t


De morgan s laws
DE MORGAN’S LAWS:

  • The negation of an and statement is logically equivalent to the or statement in which each component is negated.

  • Symbolically ~(p ∧ q) ≡ ~p ∨ ~q.

  • The negation of an or statement is logically equivalent to the and statement in which each component is negated.

  • Symbolically: ~(p ∨ q) ≡ ~p ∧ ~q.

  • ~(p ∨ q) ≡ ~p ∧ ~q



Exercise
EXERCISE:

  • (p ∧ q) ∧ r = p ∧(q ∧ r)

  • Are the statements (p∧q)∨r and p ∧ (q ∨ r) logically equivalent?


Tautology
TAUTOLOGY:

  • A tautology is a statement form that is always true regardless of the truth values of the statement variables.

  • A tautology is represented by the symbol “T”..

  • EXAMPLE: The statement form p ∨ ~ p is tautology


Contradiction
CONTRADICTION:

  • A contradiction is a statement form that is always false regardless of the truth values of the statement variables.

  • A contradiction is represented by the symbol “c”.

  • EXAMPLE: The statement form p ∧ ~ p is a contradiction.


Remarks
REMARKS:

  • Most statements are neither tautologies nor contradictions.

  • The negation of a tautology is a contradiction and vice versa.

  • In common usage we sometimes say that two statements are contradictory.

  • By this we mean that their conjunction is a contradiction: they cannot both be true.


Logical equivalence involving tautology contyradiction
LOGICAL EQUIVALENCE INVOLVING TAUTOLOGY / CONTYRADICTION

Show that p∧c = c

  • Show that p ∧ t = p



Use truth table to show that p q p q is a contradiction
Use truth table to show that (p ∧ ~q) ∧(~ is a tautologyp∨q) is a contradiction.


Exclusive or
EXCLUSIVE OR: is a tautology

  • I shall buy a pen or a book. (Inclusive)

  • Tomorrow at 9, I’ll be in Lahore or Islamabad


Applying laws of logic
APPLYING LAWS OF LOGIC is a tautology

  • Using law of logic, simplify the statement form p ∨ [~(~p ∧ q)]

  • Solution:

  • p ∨ [~(~p ∧ q)] = p ∨ [~(~p) ∨ (~q)] DeMorgan’s Law

  • =p ∨ [p∨(~q)] Double Negative Law

  • = [p ∨ p]∨(~q) Associative Law for ∨ =p ∨ (~q) IndempotentLaw


P q p q p
~ (~ p ∧ q) ∧ (p ∨ q) =p is a tautology

  • ~(~p ∧ q) ∧ (p∨q) = (~(~p) ∨ ~q) ∧(p ∨ q) DeMorgan’sLaw

  • = (p ∨ ~q) ∧ (p∨q) Double Negative Law

  • = p ∨ (~q ∧ q) Distributive Law

  • = p ∨ c Negation Law

  • = p Identity Law


Conditional statements
CONDITIONAL STATEMENTS: is a tautology

  • Consider the statement: "If you earn an A in Math, then I'll buy you a computer."

  • This statement is made up of two simpler statements

  • p: "You earn an A in Math," and q: "I will buy you a computer."

  • The original statement is then saying: if p is true, then q is true, or, more simply, if p, then q.

  • We can also phrase this as p implies q, and we write p → q.


Conditional statements or implications
CONDITIONAL STATEMENTS OR IMPLICATIONS: is a tautology

  • If p and q are statement variables, the conditional of q by p is “If p then q”

  • or “p implies q” and is denoted p → q.

  • It is false when p is true and q is false; otherwise it is true.

  • The arrow "→ " is the conditional operator, and in p → q the statement p is called the hypothesis (or antecedent) and q is called the conclusion (or consequent).



Practice with conditional statements
PRACTICE WITH CONDITIONAL STATEMENTS: is a tautology

  • 1. “If 1 = 1, then 3 = 3.” TRUE

  • 2. “If 1 = 1, then 2 = 3.” FALSE

  • 3. “If 1 = 0, then 3 = 3.” TRUE

  • 4. “If 1 = 2, then 2 = 3.” TRUE

  • 5. “If 1 = 1, then 1 = 2 and 2 = 3.” FALSE

  • 6. “If 1 = 3 or 1 = 2 then 3 = 3.” TRUE


Alternative ways of expressing implications
ALTERNATIVE WAYS OF EXPRESSING IMPLICATIONS: is a tautology

  • The implication p → q could be expressed in many alternative ways as:

  • “if p then q” •“not p unless q”

  • “p implies q” •“q follows from p”

  • “if p, q” •“q if p”

  • “p only if q” •“q whenever p”

  • “p is sufficient for q” •“q is necessary for p”


Alternative ways of expressing implications1
ALTERNATIVE WAYS OF EXPRESSING IMPLICATIONS: is a tautology

  • That you get the job implies that you have the best credentials.

    • If you get the job, then you have the best credentials

  • It is necessary to walk 8 miles to get to the top of the Peak.

    • If you get to the top of the peak, then you must have walked 8 miles.


Hierarchy of operations for logical connectives
HIERARCHY OF OPERATIONS FOR LOGICAL CONNECTIVES is a tautology

  • ~ (negation)

  • ∧ (conjunction), ∨ (disjunction)

  • → (conditional)


Translating english sentences to symbols
TRANSLATING ENGLISH SENTENCES TO SYMBOLS: is a tautology

  • Let p and q be propositions:

  • p = “you get an A on the final exam”

  • q = “you do every exercise in this book”

  • r = “you get an A in this class”

  • To get an A in this class it is necessary for you to get an A on the final. p → r

  • You do every exercise in this book; You get an A on the final, implies, you get an A in the class.

  • p ∧ q → r


Construct a truth table for the statement form p q p
Construct a truth table for the statement form p is a tautology ∨ ~ q → ~ p


Logical equivalence involving implication
LOGICAL EQUIVALENCE INVOLVING IMPLICATION is a tautology

  • Use truth table to show p→q = ~q → ~p


Implication law p q p q
IMPLICATION LAW is a tautologyp→q = ~p∨q


Negation of a conditional statement
NEGATION OF A CONDITIONAL STATEMENT: is a tautology

  • Since p→q = ~p∨q therefore

  • ~ (p → q) = ~ (~ p ∨ q) = ~ (~ p) ∧ (~ q) by De Morgan’s law

  • = p ∧ ~ q by the Double Negative law

  • Thus the negation of “if p then q” is logically equivalent to “p and not q”.

  • Accordingly, the negation of an if-then statement does not start with the word if.


Examples
EXAMPLES is a tautology

  • Write negations of each of the following statements:

  • 1. If Ali lives in Pakistan then he lives in Lahore.

  • 2. If my car is in the repair shop, then I cannot get to class. 3.

  • If x is prime then x is odd or x is 2.

  • 4. If n is divisible by 6, then n is divisible by 2 and n is divisible by 3.


Solutions
SOLUTIONS: is a tautology

  • 1. Ali lives in Pakistan and he does not live in Lahore.

  • 2. My car is in the repair shop and I can get to class.

  • 3. x is prime but x is not odd and x is not 2.

  • 4. n is divisible by 6 but n is not divisible by 2 or by 3


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