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Introduction To Logic Essay

2695 words - 11 pages

Predicate Logic

Example:
All men are mortal.
Socrates is a man.
··· Socrates is mortal.
Note: We need logic laws that work for statements involving quantities like “some” and “all”.
In English, the predicate is the part of the sentence that tells you
something about the subject.

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More on predicates
Example: Nate is a student at UT.
What is the subject? What is the predicate?
Example: We can form two different predicates.
Let P(x) be “x is a student at UT”.
Let Q(x, y) be “x is a student at y”.
Definition: A predicate is a property that a variable or a finite
collection of variables can have. A predicate becomes a proposition
when specific values are assigned to the variables. P(x1, x2, ..., ...view middle of the document...

Ways to read ∀xP (x):
For every x, P(x)
For every x, P(x) is true
For all x, P(x)

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More on the universal quantifier
Definition: A counterexample for ∀xP (x) is any t ∈ U , where
U is the universe, such that P(t) is false.
Some Examples
Example: P(x, y): x + y = 8
Assign x to be 1, and y to be 7. We get proposition P(1, 7) which is
true.
Proposition P(2, 5) is false since 2 + 5 = 8.
Example: ∀x[x ≥ 0]
U = N (non-negative integers)
We could re-write this proposition as: ∀x ∈ N, x ≥ 0
Is the proposition true?
What if the universe is R?
Example: ∀x∀y[x + y > x]
Is this proposition true if:
1. If U = N?
2. If U = R?
Example: ∀x∀y[x > y]
True if:
universe for x = the non-negative integers
universe for y = the non-positive integers

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The Existential Quantifier: ∃
Definition: The symbol ∃ is call the existential quantifier and
represents the phrase “there exists” or “for some”. The existential
quantification of P(x) is the statement “P(x) for some values x in
the universe”, or equivalently, “There exists a value for x such that
P(x) is true”, which is written ∃xP (x).
Note: If P(x) is true for at least one element in the domain, then
∃xP (x) is true. Otherwise it is false.
Note: Let P(x) be a predicate and c ∈ U (U = domain).
The following implications are true:
∀xP (x) → P (c)
P (c) → ∃xP (x)
Example: ∃x [x is prime] where U = Z
Is this proposition true or false?
Example: ∃x[x2 < 0] where U = R
True or false?
Exercises: True or false? Prove your answer.
1. ∃n[n2 = n] where U = Z.
2. ∃n[n2 = n] where U = {4, 5, 6, 7}.

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Translating Quantified Statements
Translate the following into English.
1. ∀x[x2 ≥ 0] where U = R.
2. ∃t[(t > 3) ∧ (t3 > 27)] where U = R.
3. ∀x[(2|x) ∨ (2 |x)] where U = N
Translate the following into logic statements.
1. There is an integer whose square is twice itself.
2. No school buses are purple.
3. If a real number is even, then its square is even.
Note: Let U = {1, 2, 3}.
Proposition ∀xP (x) is equivalent to P (1) ∧ P (2) ∧ P (3).
Proposition ∃xP (x) is equivalent to P (1) ∨ P (2) ∨ P (3).

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Bound and Free Variables
Definition: All variables in a predicate must be bound to turn a
predicate into a proposition. We bind a variable by assigning it a
value or quantifying it. Variables which are not bound are free.
Note: If we bind one variable in a predicate P (x, y, z) with 3
variables, say by setting z = 4, we get a predicate with 2 variables:
P (x, y, 4).
Example: Let U = N.
P (x, y, z) : x + y = z ← 3 free variables
Let Q(y, z) = P (2, y, z) : 2 + y = z ← 2 free variables

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Examples with Quantifiers
Example: U = Z
N(x): x is a non-negative integer
E(x): x is even
O(x): x is odd
P(x): x is prime
Translate into logical notation.
1. There exists an even integer.
2. Every integer is even or odd.
3. All prime integers are non-negative.
4. The only even prime is 2.
5. Not all integers are odd.
6. Not all primes are odd.
7. If an integer is not odd, then it is even.

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Examples with Nested Quantifiers
Note about nested...

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