# Let’s Talk About Logic - PowerPoint PPT Presentation

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Let’s Talk About Logic. Logic is a system based on propositions . A proposition is a statement that is either true or false (not both). We say that the truth value of a proposition is either true (T) or false (F). Corresponds to 1 and 0 in digital circuits.

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#### Presentation Transcript

• Logic is a system based on propositions.

• A proposition is a statement that is either true or false (not both).

• We say that the truth value of a proposition is either true (T) or false (F).

• Corresponds to 1 and 0 in digital circuits

Applied Discrete Mathematics

Week 1: Logic and Sets

### Logical Operators (Connectives)

• Negation (NOT)

• Conjunction (AND)

• Disjunction (OR)

• Exclusive or (XOR)

• Implication (if – then)

• Biconditional (if and only if)

• Truth tables can be used to show how these operators can combine propositions to compound propositions.

Applied Discrete Mathematics

Week 1: Logic and Sets

• A tautology is a statement that is always true.

• Examples:

• R(R)

• (PQ)(P)(Q)

• If ST is a tautology, we write ST.

• If ST is a tautology, we write ST.

Applied Discrete Mathematics

Week 1: Logic and Sets

• A contradiction is a statement that is alwaysfalse.

• Examples:

• R(R)

• ((PQ)(P)(Q))

• The negation of any tautology is a contradiction, and the negation of any contradiction is a tautology.

Applied Discrete Mathematics

Week 1: Logic and Sets

### Propositional Functions

• Propositional function (open sentence):

• statement involving one or more variables,

• e.g.: x-3 > 5.

• Let us call this propositional function P(x), where P is the predicate and x is the variable.

What is the truth value of P(2) ?

false

What is the truth value of P(8) ?

false

What is the truth value of P(9) ?

true

Applied Discrete Mathematics

Week 1: Logic and Sets

### Propositional Functions

• Let us consider the propositional function Q(x, y, z) defined as:

• x + y = z.

• Here, Q is the predicate and x, y, and z are the variables.

What is the truth value of Q(2, 3, 5) ?

true

What is the truth value of Q(0, 1, 2) ?

false

What is the truth value of Q(9, -9, 0) ?

true

Applied Discrete Mathematics

Week 1: Logic and Sets

### Universal Quantification

• Let P(x) be a propositional function.

• Universally quantified sentence:

• For all x in the universe of discourse P(x) is true.

• Using the universal quantifier :

• x P(x) “for all x P(x)” or “for every x P(x)”

• (Note: x P(x) is either true or false, so it is a proposition, not a propositional function.)

Applied Discrete Mathematics

Week 1: Logic and Sets

### Universal Quantification

• Example:

• S(x): x is a UMB student.

• G(x): x is a genius.

• What does x (S(x)  G(x)) mean ?

• “If x is a UMB student, then x is a genius.”

• or

• “All UMB students are geniuses.”

Applied Discrete Mathematics

Week 1: Logic and Sets

### Existential Quantification

• Existentially quantified sentence:

• There exists an x in the universe of discourse for which P(x) is true.

• Using the existential quantifier :

• x P(x) “There is an x such that P(x).”

• “There is at least one x such that P(x).”

• (Note: x P(x) is either true or false, so it is a proposition, but no propositional function.)

Applied Discrete Mathematics

Week 1: Logic and Sets

### Existential Quantification

• Example:

• P(x): x is a UMB professor.

• G(x): x is a genius.

• What does x (P(x)  G(x)) mean ?

• “There is an x such that x is a UMB professor and x is a genius.”

• or

• “At least one UMB professor is a genius.”

Applied Discrete Mathematics

Week 1: Logic and Sets

### Quantification

• Another example:

• Let the universe of discourse be the real numbers.

• What does xy (x + y = 320) mean ?

• “For every x there exists a y so that x + y = 320.”

Is it true?

yes

Is it true for the natural numbers?

no

Applied Discrete Mathematics

Week 1: Logic and Sets

### Disproof by Counterexample

• A counterexample to x P(x) is an object c so that P(c) is false.

• Statements such as x (P(x)  Q(x)) can be disproved by simply providing a counterexample.

Statement: “All birds can fly.”

Disproved by counterexample: Penguin.

Applied Discrete Mathematics

Week 1: Logic and Sets

### Negation

• (x P(x)) is logically equivalent to x (P(x)).

• (x P(x)) is logically equivalent to x (P(x)).

• See Table 3 in Section 1.3.

• I recommend exercises 5 and 9 in Section 1.3.

Applied Discrete Mathematics

Week 1: Logic and Sets

### … and now for something completely different…

• Set Theory

Actually, you will see that logic and set theory are very closely related.

Applied Discrete Mathematics

Week 1: Logic and Sets

### Set Theory

• Set: Collection of objects (“elements”)

• aA “a is an element of A” “a is a member of A”

• aA “a is not an element of A”

• A = {a1, a2, …, an} “A contains…”

• Order of elements is meaningless

• It does not matter how often the same element is listed.

Applied Discrete Mathematics

Week 1: Logic and Sets

### Set Equality

• Sets A and B are equal if and only if they contain exactly the same elements.

• Examples:

• A = {9, 2, 7, -3}, B = {7, 9, -3, 2} :

A = B

• A = {dog, cat, horse}, B = {cat, horse, squirrel, dog} :

A  B

• A = {dog, cat, horse}, B = {cat, horse, dog, dog} :

A = B

Applied Discrete Mathematics

Week 1: Logic and Sets

### Examples for Sets

• “Standard” Sets:

• Natural numbers N = {0, 1, 2, 3, …}

• Integers Z = {…, -2, -1, 0, 1, 2, …}

• Positive Integers Z+ = {1, 2, 3, 4, …}

• Real Numbers R = {47.3, -12, , …}

• Rational Numbers Q = {1.5, 2.6, -3.8, 15, …}(correct definition will follow)

Applied Discrete Mathematics

Week 1: Logic and Sets

### Examples for Sets

• A = “empty set/null set”

• A = {z} Note: zA, but z  {z}

• A = {{b, c}, {c, x, d}}

• A = {{x, y}} Note: {x, y} A, but {x, y}  {{x, y}}

• A = {x | P(x)}“set of all x such that P(x)”

• A = {x | xN x > 7} = {8, 9, 10, …}“set builder notation”

Applied Discrete Mathematics

Week 1: Logic and Sets

### Examples for Sets

• We are now able to define the set of rational numbers Q:

• Q = {a/b | aZ bZ+}

• or

• Q = {a/b | aZ bZ  b0}

• And how about the set of real numbers R?

• R = {r | r is a real number}That is the best we can do.

Applied Discrete Mathematics

Week 1: Logic and Sets

### Subsets

• A  B “A is a subset of B”

• A  B if and only if every element of A is also an element of B.

• We can completely formalize this:

• A  B  x (xA  xB)

• Examples:

A = {3, 9}, B = {5, 9, 1, 3}, A  B ?

true

A = {3, 3, 3, 9}, B = {5, 9, 1, 3}, A  B ?

true

A = {1, 2, 3}, B = {2, 3, 4}, A  B ?

false

Applied Discrete Mathematics

Week 1: Logic and Sets

U

B

C

A

### Subsets

• Useful rules:

• A = B  (A  B)  (B  A)

• (A  B)  (B  C)  A  C (see Venn Diagram)

Applied Discrete Mathematics

Week 1: Logic and Sets

### Subsets

• Useful rules:

•   A for any set A

• A  A for any set A

• Proper subsets:

• A  B “A is a proper subset of B”

• A  B  x (xA  xB)  x (xB  xA)

• or

• A  B  x (xA  xB)  x (xB  xA)

Applied Discrete Mathematics

Week 1: Logic and Sets

### Cardinality of Sets

• If a set S contains n distinct elements, nN,we call S a finite set with cardinality n.

• Examples:

• A = {Mercedes, BMW, Porsche}, |A| = 3

B = {1, {2, 3}, {4, 5}, 6}

|B| = 4

C = 

|C| = 0

D = { xN | x  7000 }

|D| = 7001

E = { xN | x  7000 }

E is infinite!

Applied Discrete Mathematics

Week 1: Logic and Sets

### The Power Set

• 2Aor P(A) “power set of A”

• 2A = {B | B  A} (contains all subsets of A)

• Examples:

• A = {x, y, z}

• 2A = {, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z}}

• A = 

• 2A = {}

• Note: |A| = 0, |2A| = 1

Applied Discrete Mathematics

Week 1: Logic and Sets

### The Power Set

• Cardinality of power sets:

• | 2A | = 2|A|

• Imagine each element in A has an “on/off” switch

• Each possible switch configuration in A corresponds to one element in 2A

• For 3 elements in A, there are 222 = 8 elements in 2A

Applied Discrete Mathematics

Week 1: Logic and Sets

### Cartesian Product

• The ordered n-tuple (a1, a2, a3, …, an) is an ordered collection of objects.

• Two ordered n-tuples (a1, a2, a3, …, an) and (b1, b2, b3, …, bn) are equal if and only if they contain exactly the same elements in the same order, i.e. ai = bi for 1  i  n.

• The Cartesian product of two sets is defined as:

• AB = {(a, b) | aA  bB}

• Example: A = {x, y}, B = {a, b, c}AB = {(x, a), (x, b), (x, c), (y, a), (y, b), (y, c)}

Applied Discrete Mathematics

Week 1: Logic and Sets

### Cartesian Product

• Note that:

• A = 

• A = 

• For non-empty sets A and B: AB  AB  BA

• |AB| = |A||B|

• The Cartesian product of two or more sets is defined as:

• A1A2…An = {(a1, a2, …, an) | aiA for 1  i  n}

Applied Discrete Mathematics

Week 1: Logic and Sets

### Set Operations

• Union: AB = {x | xA  xB}

• Example: A = {a, b}, B = {b, c, d}

• AB = {a, b, c, d}

• Intersection: AB = {x | xA  xB}

• Example: A = {a, b}, B = {b, c, d}

• AB = {b}

Applied Discrete Mathematics

Week 1: Logic and Sets