Tautologies and Contradictions


















What's a Tautology?

	A logical expression that is always true
	(no matter what values are assigned to the variables)

What's a Contradiction?

	A logical expression that is always false
	(no matter what values are assigned to the variables)

Use a truth table to determine tautology or contradiction.

	P or not P

	P and not P

	P or (P and Q)	<->  P


















Classwork
You may work with a partner.

Use a truth table to determine tautology or contradiction.

	P -> Q 	<->  not P or Q


















Logical Equivalences (Tautologies) (page 29)

	P and True	<=>  P			Identity
	P or False	<=>  P			Identity
	P or True	<=>  True		Domination
	P and False	<=>  False		Domination
	P or P		<=>  P			Idempotent
	P and P		<=>  P			Idempotent
	not (not P)	<=>  P			Double Negation
	P or Q	 	<=>  Q or P		Commutative
	P and Q		<=>  Q and P		Commutative
	(P or Q) or R	<=>  P or (Q or R)	Associative
	(P and Q) and R	<=>  P and (Q and R)	Associative
	P or (Q and R)	<=>  (PvQ) and (PvR)	Distributive
	P and (Q or R)	<=>  (P^Q) or (P^R)	Distributive
	not (P and Q)	<=>  not P or not Q	DeMorgan's
	not (P or Q) 	<=>  not P and not Q	DeMorgan's
	P or (P and Q)	<=>  P			Absorption
	P and (P or Q)	<=>  P			Absorption
	P or not P	<=>  True		Negation
	P and not P	<=>  False		Negation

	P -> Q  	<=>  not P or Q
	P <-> Q 	<=>  (P -> Q) and (Q -> P)
	P <-> Q 	<=>  (P and Q) or (not P and not Q)


Using the Laws

Suppose you replace P with (A -> B) in a law such as (P or P) <=> P.
What do you know about the resulting expression?

	((A -> B) or (A -> B)) <=> (A -> B)

Suppose you have the following law and expression.
How do you apply the law to the expression?

	law: not (P and Q) <=> not P or not Q

	exp: not (P and Q) or P


















Use algebra to simplify the code.


	if (outflow > inflow && !(outflow > inflow && pressure < 10))
	    do something;
	else
	    do something else;


















Classwork
You may work with a partner.

Use algebra to simplify the code.


	if ( !(v1 < v2 || odd(n)) || !(v1 < v2) && odd(n) )
	   statement1;
	else
	   statement2;


















Another Look at Distributive Laws

Does multiply distribute over add?

	a * (b + c) = a*b + a*c

Does AND distribute over OR?

	P and (Q or R) <=> (P and Q) or (P and R)

Does add distribute over multiply?

	a + (b * c) = (a+b) * (a+c)

Does OR distribute over AND?

	P or (Q and R) <=> (P or Q) and (P or R)