Relational Calculus [194]

Propositional Calculus

• TRUE
• FALSE
• AND ($\wedge$)
• NOT ($\neg$)
• IMPLIES ( $p \Rightarrow q$ means if $p$ then $q$)

Predicate Calculus adds quantification

• Universal Quantification: FORALL ($\forall$)
• Existential Quantification: EXISTS ($\exists$)

Tuple Relational Calculus adds

• an actual database or set of relations

Propositional Calculus Rules [195]

• $a \wedge 1 = a$
• $a \vee 0 = a$
• $a \wedge \neg a = 0$
• $a \wedge 0 = 0$
• $a \vee \neg a = 1$
• $a \vee 1 = 1$
• $a \wedge a = a$
• $a \vee a = a$
• $a \wedge b = b \wedge a$
• $a \vee b = b \vee a$

• $a \wedge (b \wedge c) = (a \wedge b) \wedge c$
• $a \vee (b \vee c) = (a \vee b) \vee c$
• $a \wedge (b \vee c) = (a \wedge b) \vee (a \wedge c)$
• $a \vee (b \wedge c) = (a \vee b) \wedge (a \vee c)$
• $\neg \neg a = a$
• $\neg (a \wedge b) = \neg a \vee \neg b$
• $\neg (a \vee b) = \neg a \wedge \neg b$

Implication [196]

$A$ Implies $B$

• $A \Rightarrow B$
• if $A$ then $B$
• if $A$ is not true, does not say anything about $B$

a	b	a $\Rightarrow$ b
1	1	1
0	1	1
1	0	0
0	0	1

$a \Rightarrow b \equiv \neg a \vee b$

A formula may be

• a tautology - $a \vee \neg a$ is always true
• satisfiable - $a \Rightarrow b$ may be true
• unsatisfiable - $a \wedge \neg a$ is never true

Predicate Calculus [197]

Predicate Calculus or First Order Logic adds

• $\exists$ - there exists, the Existential Quantifier
• $\forall$ - for all, the Universal Quantifier
• predicates based on membership in a relation

Examples of Predicates

• apple(x) says that x is an apple
• or that x is a member of the APPLE relation
• red(x), person(x), motherof(x,y)

Examples of $\exists$

• $\exists x (apple(x) \wedge red(x))$
• $\exists x \exists y (person(x) \wedge person(y) \wedge motherof(x,y))$
• $\exists x \exists y (person(x) \wedge person(y) \wedge loves(x,y))$

Universal Quantification [198]

• $\forall x (apple(x) \Rightarrow red(x))$
• $\forall x (person(x) \Rightarrow \exists y(person(y) \wedge motherof(y,x)))$
• $\forall x (person(x) \Rightarrow \exists y(person(y) \wedge motherof(x,y)))$
• $\forall x (mother(x) \Rightarrow female(x))$
• $\forall x (person(x) \Rightarrow \exists y(person(y) \wedge loves(x,y)))$
• $\forall x (person(x) \Rightarrow \exists t \exists y (person(y) \wedge time(t) \wedge$
• $lovesat(x,y,t)))$
• note that $\forall x \equiv \neg \exists x \neg$
• $\forall x(x \Rightarrow y)\equiv \neg \exists x \neg (x \Rightarrow y) \equiv \... ...exists x \neg (\neg x \vee y) \equiv \neg \exists x (\neg \neg x \wedge \neg y)$
• $\equiv \neg \exists x (x \wedge \neg y)$
• $\forall x (apple(x) \Rightarrow red(x)) \equiv \neg \exists x(apple(x) \wedge \neg red(x))$
• Lincoln: `You can fool some of the people all of the time'
• $\exists x(person(x) \wedge (\forall y(time(y) \Rightarrow foolat(x,y))))$
• `But you cannot fool all the people all the time'
• $\exists x \exists y (p(x) \wedge t(y) \wedge \neg f(x,y))$