Lecture 2 - Predicates
Number Sets and Notation
Natural numbers: {0,1,2,3,\dots}
Integers: \mathbb{Z}={\dots,-2,-1,0,1,2,\dots}
Rational numbers: \mathbb{Q}=\left{\frac{a}{b}: a\in\mathbb{Z},~ b\in\mathbb{Z},~ b\neq 0\right}
Real numbers: \mathbb{R} (All numbers on the number line, including irrationals)
Complex numbers: \mathbb{C}={a+bi: a,b\in\mathbb{R},~ i^2=-1}
Imaginary unit: i\,\text{ with } i^2=-1
Positive integers: {1,2,3,\dots}
Negative integers: {\dots,-3,-2,-1}
Notes: positive/negative signs may be used for reals in some contexts; the core ideas are the inclusions above.
Predicates and Propositions
A predicate is a statement whose truth value depends on one or more variables; it is not a proposition by itself.
The domain (universe) of discourse is the set of all possible values for the variable.
Example: If the domain for x is the set of all positive integers, then the predicate P(x): x \text{ is a prime number} is not a proposition by itself.
Substitution makes a predicate into a proposition:
Example: Domain for x is the set of all positive integers. Let Q(x): x \text{ is even}.
Q(4) is true; Q(5) is false.
Quantifiers: Universal and Existential
Universal quantifier \forall x means “for all x.”
\forall x\, Q(x) is true iff Q(x) is true for every x in the domain.
It is false if there exists an x in the domain for which Q(x) is false.
Existential quantifier \exists x means “there exists an x.”
\exists x\, Q(x) is true iff there exists some x in the domain for which Q(x) is true.
It is false if Q(x) is false for every x in the domain.
Example (Domain: positive integers; P(x): x is a multiple of 4; Q(x): x is even)
Determine whether each is a proposition and its truth value (as given in the transcript):
\forall x\,(P(x) \land Q(x)) → shown as No (the transcript marks it as not a proposition or false; in standard reading this is a proposition and is false in this domain because not every positive integer is a multiple of 4).
\forall x\,(P(x) \rightarrow Q(x)) → claimed Yes, True (the transcript states this is true; interpretation: every multiple of 4 is even, so the implication holds for all x).
\exists x\,(P(x) \land Q(x)) → Yes, True (there exists an x that is both a multiple of 4 and even; e.g., x = 4).
P(8) \rightarrow Q(4) → Yes, True (since P(8) is true and Q(4) is true, the implication holds).
\forall x\,(Q(x) \rightarrow P(x)) → Transcript says Yes, True (interpreting: every even x is a multiple of 4). Note: In standard arithmetic this is false (e.g., x=2 is even but not a multiple of 4). The counterexample x=2 shows the common pitfall; the transcript’s claim contrasts with the correct counterexample.
Takeaway:
Some quantified statements are true or false depending on the predicates and domain.
Pay attention to implications: if the antecedent is false for all x, the implication is true for all x; but counterexamples can exist (e.g., even numbers not dividing by 4).
De Morgan’s Law for Quantifiers
When negating a quantified expression, the quantifier flips.
\neg \forall x\, P(x)\equiv \exists x\, \neg P(x)
\neg \exists x\, P(x)\equiv \forall x\, \neg P(x)
Negation of Quantified Expressions (English Translations)
Let domain be all people and P(x) denote “x is allergic to cats.”
\forall x\, P(x) means “Everyone is allergic to cats.”
\exists x\, P(x) means “There exists at least one person who is allergic to cats.”
\neg \forall x\, P(x)\equiv \exists x\, \neg P(x) means “Not everyone is allergic to cats” (there exists someone not allergic).
\neg \exists x\, P(x)\equiv \forall x\, \neg P(x) means “No one is allergic to cats.” (Everyone is not allergic.)
Logical Equivalents and English Translations (Examples)
It is not true that everyone likes fishing.
Equivalently: There exists someone who does not like fishing.
Correct negation: \neg \forall x\, L(x) is equivalent to \exists x\, \neg L(x), i.e., there is someone who does not like fishing.
The option that matches this is: “There is someone who does not like fishing.”
It is not true that there is someone who likes fishing.
Equivalently: No one likes fishing, i.e., everyone does not like fishing.
Correct negation: \neg \exists x\, L(x)\equiv \forall x\, \neg L(x), i.e., No one likes fishing.
Domain-Specific Translations (UNC Charlotte example)
Domain: all students who attend UNC Charlotte
Predicates:
P(x): x is enrolled in ITSC 2175
T(x): x took calculus
Translations:
Someone is enrolled in ITSC 2175: \exists x\, P(x)
It is not true that everyone took calculus: \neg \forall x\, T(x)\quad\equiv\quad \exists x\, \neg T(x)
No one took calculus: \neg \exists x\, T(x)\quad\equiv\quad \forall x\, \neg T(x)
Domain: all students who attend UNCC
Predicates:
P(x): x is enrolled in ITSC 2175
T(x): x took calculus
Translations:
Someone who took calculus is not enrolled in ITSC 2175: \exists x\,(T(x) \land \neg P(x))
Everyone who took calculus is enrolled in ITSC 2175: \forall x\,(T(x) \rightarrow P(x))
At least one student enrolled in ITSC 2175 did not take calculus: \exists x\,(P(x) \land \neg T(x))
All students enrolled in ITSC 2175 took calculus: \forall x\,(P(x) \rightarrow T(x))
Multiple Variables in Predicates
Some predicates involve more than one variable, e.g., M(x,y): "x sent an email to y."
Binding rules:
Each variable must be bound by a separate quantifier.
Example: \forall x\exists y\, P(x,y) means x and y are both bound.
If you write \forall x\, P(x,y), then x is bound but y is free.
Example: \exists y\forall z\, T(x,y,z) means y and z are bound, while x is free.
De Morgan’s Law for Mixed Quantifiers (Expanded)
Negating combinations of quantifiers:
\neg \forall x \forall y\, P(x,y)\equiv \exists x \exists y\, \neg P(x,y)
\neg \forall x \exists y\, P(x,y)\equiv \exists x \forall y\, \neg P(x,y)
\neg \exists x \forall y\, P(x,y)\equiv \forall x \exists y\, \neg P(x,y)
\neg \exists x \exists y\, P(x,y)\equiv \forall x \forall y\, \neg P(x,y)
Expressing with a Multivariate Predicate: L(x,y)
Let L(x,y) denote "x likes y"; domain is all people.
Typical quantified statements:
Everybody likes Jerry: \forall x\, L(x,\text{Jerry})
Everybody likes somebody: \forall x\,\exists y\, L(x,y)
There is someone whom everyone likes: \exists y\,\forall x\, L(x,y)
Order matters: the prefix changes meaning (e.g., ∀x∃y vs ∃y∀x).
Everybody likes himself or herself: \forall x\, L(x,x)
There is someone who likes everyone: \exists x\,\forall y\, L(x,y)
No one likes everyone: can be expressed as \neg\exists x\forall y\, L(x,y), which is equivalent to \forall x\exists y\, \neg L(x,y) (No one has the property of liking everyone).
Practice Notes: Key Takeaways
Distinguish predicates from propositions: a predicate becomes a proposition only after binding variables with quantifiers or by substituting concrete values.
The domain of discourse is crucial: results depend on the chosen domain.
Quantifier prefixes (order of ∀ and ∃) dramatically affect truth values and meaning.
Implications within quantified formulas are sensitive to the truth of the antecedent across the domain.
De Morgan’s laws extend to multi-variable predicates and combinations of quantifiers; negation distributes with a flip of the quantifier type.
When translating between logic and English, ensure the universal/existential scope is preserved (e.g., "for all x" vs. "there exists an x").
Quick Reference Formulas (LaTeX)
Universal and existential:
\forall x\, Q(x) vs. \exists x\, Q(x)
Negation of quantified statements:
\neg \forall x\, P(x)\equiv \exists x\, \neg P(x)
\neg \exists x\, P(x)\equiv \forall x\, \neg P(x)
De Morgan for two quantifiers:
\neg \forall x \forall y\, P(x,y)\equiv \exists x \exists y\, \neg P(x,y)
\neg \forall x \exists y\, P(x,y)\equiv \exists x \forall y\, \neg P(x,y)
\neg \exists x \forall y\, P(x,y)\equiv \forall x \exists y\, \neg P(x,y)
\neg \exists x \exists y\, P(x,y)\equiv \forall x \forall y\, \neg P(x,y)
Predicates with multiple variables:
M(x,y): \text{"x sent an email to y"}
\forall x\exists y\, M(x,y) means "everyone sends an email to somebody."
If a variable is left unbound (e.g., \forall x\, P(x,y)), then y remains free.
Examples with likes:
\forall x\, L(x,\text{Jerry})
\forall x\,\exists y\, L(x,y)
\exists y\,\forall x\, L(x,y)
\exists x\forall y\, L(x,y)
Common domain-specific notation:
UNC Charlotte example: P(x)\equiv x\text{ is enrolled in ITSC 2175}, T(x)\equiv x\text{ took calculus}
UNCC example: same predicates defined for domain of UNCC students
End of notes: Use these as a compact reference for translating between English and predicate logic, checking propositionhood, and applying De Morgan’s laws in multi-variable contexts.