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Discrete StructuresCMSC 250Lecture 1

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Administrative Syllabus Lecture section: MWF 10-11 Lab/discussion section Every week (mostly):

– Worksheet– Quiz – Homework

Class webpage Three midterms, as noted on the syllabus (on the class

webpage) Academic integrity

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Motivation

Why learn this material? Some things can be directly applied Some things are good to know Some things just teach a way of thinking and

expressing yourself

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Examples

Directly applied- circuits to do addition Conditionals in if statements Things which are good to know- for example that the

square root of 2 can’t be stored Way of thinking- the number of primes is infinite

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Overall theme

Proofs of theorems Examples of theorems

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Course topics Propositional logic (and circuits) Predicate calculus - quantification Elementary number theory Mathematical induction Sets Counting - combinations and probability Functions Relations Graph theory

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Chapter 1, Propositional Logic

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Statement/proposition

Declarative (true or false) Symbolized by a letter Examples:

– Today is Monday.– 5 + 2 = 7– 3 * 6 > 18 – The sky is blue.– Why is the sky blue?– Barack Obama.– Two students in the class have a GPA of 3.275.– This sentence is false.– The current king of France is bald.

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Other symbols and definitionsfor constructing compound statements

Conjunction"and" is symbolized by

Disjunction"or" is symbolized by

Negation"not" is symbolized by ~ (or sometimes )

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Truth table for

p q p q

1 1

1 0

0 1

0 0

1

0

0

0

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Truth table for

p q p q

1 1

1 0

0 1

0 0

1

1

1

0

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Truth table for

p p

1

0

0

1

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Translation of English to symbolic logic statements

The sky is blue.– one simple (atomic) statement- assign a letter or name to it, i.e., b

The sky is blue and the grass is green.– one statement– conjunction of two atomic statements– each single statement gets a letter or name, i.e., b, g– and join with ^ i.e., b ^ g

The sky is blue or the sky is purple.– one statement– disjunction of two atomic statements– each single statement gets a name, i.e., b, p– and join with i.e., b p

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Translation #1 The sky is blue or purple.

– two statements• "the sky is blue"- name this b• "the sky is purple"- name this p

– it's still a disjunction• "the sky is blue" or "the sky is purple"• b p

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Translation #2

The sky is blue but not dark.– two statements

• "the sky is blue"- name this b• "the sky is dark"- name this d

– conjunction with negation• "the sky is blue" and "the sky is not dark"• the sky is blue and it is not the case that the sky is dark • "it is not the case that the sky is dark" is ~d• b ^ ~ d

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Translation #3

2 x 6 English: x is greater than or equal to 2 and less than

or equal to 6 Two statements:

– "x is greater than or equal to 2"- name this p– "x is less than or equal to 6"- name this q

Becomes p ^ q

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#3 continued- 2 x 6

p is actually a compound statement– x is greater than 2 or x is equal to 2 r s– "x is greater than 2" is symbolized by r– "x is equal to 2" is symbolized by s

q is actually a compound statement– x is less than 6 or x is equal to 6 m n– "x is less than 6" is symbolized by m– "x is equal to 6" is symbolized by n

p ^ q becomes (r s) ^ (m n)

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The exclusive or

The exclusive or of p and q is p or q, but not both It’s indicated using It’s the same as (p q) (p q)

p q p q

1 1

1 0

0 1

0 0

0

1

1

0

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Precedence between the operators

~ (not) has highest precedence ^ (and) / (or) have equal precedence use parentheses to override the operators' default

precedence, or when a statement has operators with the same precedence, or just for clarity

a ^ b c

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Other "more advanced" truth tables

(p ^ r) (q ^ r)

(p ^ ~r) (p ^ r)

(p ^ q) (~q ~p)

(p ^ ~r) (q ~r)

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Discrete StructuresCMSC 250Lecture 2

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Special truth tables

p ~p

p ^ ~p

(p ^ ~q) vs. (~q ^ p)

(p ^ q) vs. ~ (~q ~p)

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Special results in the truth table

Tautological proposition– a tautology is a statement that can never be false– all of the lines of the truth table have the result "true"

Contradictory proposition– a contradiction is a statement that can never be true– all of the lines of the truth table have the result "false"

Logical equivalence of two propositions– two statements are logically equivalent if they will be true in

exactly the same cases and false in exactly the same cases– all of the lines of one column of the truth table have all of the

same truth values as the corresponding lines from another column of the truth table

– it's indicated using

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Logical equivalences Theorem 1.1.1 - page 14

Double negative:~(~p) p

Commutative:

p q q p and p ^ q q ^ p Associative:

(p q) r p (q r) and (p ^ q) ^ r p ^ (q ^ r)

Distributive:p ^ (q r) (p ^ q) (p ^ r) and p (q ^ r) (p q) ^ (p r)

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Further logical equivalences

Idempotent:

p ^ p p and p p p

Absorption: p (p ^ q) p and p ^ (p q) p

Identity:

p ^ t p and p c p

Negation:

p ~p t and p ^ ~p c Universal bound:

p ^ c c and p t t

Negations of t and c:~t c and ~c t

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DeMorgan's laws

~(p q) ~p ^ ~q– Example in English: It is not the case that Pete or Quincy went

to the store. Pete did not go to the store and Quincy did not go to the store.

~(p ^ q) ~p ~q– Example in English: It is not the case that both Pete and

Quincy went to the store. Pete did not go to the store or Quincy did not go to the store.

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Prove by truth table and by rules

~(p ~q) (~q ^ ~p) ~p

~((~p ^ q) (~p ^ ~q)) (p ^ q) p

(p q) ^ ~(p ^ q) (p ^ ~q) (q ^ ~p)

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Discrete StructuresCMSC 250Lecture 3

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Conditional statements

Hypothesis conclusion If this, then that. The hypothesis implies the conclusion. has lowest precedence (~ / ^ / ) Examples in English:

– If it is raining, I will carry my umbrella.– If you don’t eat your dinner, you will not get dessert.

p q p q

1 1

1 0

0 1

0 0

1

0

1

1

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Converting to

p q ~p q

Show with truth table

~(p q) p ^ ~q

Show with truth table and rules

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Contrapositive

Given a conditional statement p q, form its contrapositive by:– negating the conclusion and negating the hypothesis

– using the negated conclusion as the new hypothesis and the

negated hypothesis as the conclusion, e.g., ~q ~p p q ~q ~p Examples in English:

– "If Paula is here, then Quincy is here."– The contrapositive is "If Quincy is not here, then Paula is not

here."

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Converse and inverse

Given p q, for example, "If Paula is here, then Quincy is here."

To form its converse: – swap the hypothesis and the conclusion– q p– If Quincy is here, then Paula is here.

To form its inverse: – negate the hypothesis and negate the conclusion– ~p ~q– If Paula is not here, then Quincy is not here.

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Only if Translation to if-then form

– p only if q– p can be true only if q is true– if q is not true then p can't be true – if not q then not p (~q ~p)– if p then q (p q)

Translation in English– You will graduate is CS only if you pass this course.

• G only if P– If you do not pass this course then you will not graduate in CS.

• ~P ~G– If you graduate in CS then you passed this course.

• G P

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Biconditional

p if and only if q p q p iff q

p q p q

1 1

1 0

0 1

0 0

1

0

0

1

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Other English words for implication

Sufficient condition– "if r, then s" r s– means r is a sufficient condition for s– the truth of r is sufficient to ensure the truth of s

Necessary condition– "if not x, then not y" ~x ~y– means x is a necessary condition for y– the truth of x is necessary if y is true

Sufficient and necessary condition– p if, and only if q p q– the truth of p is enough to ensure the truth of q and vice versa

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Argument A sequence of statements where

– the last in the sequence is the conclusion– all others are premises (assumptions, hypotheses)

(premise1 ^ premise2 ^ …premisen) conclusion The critical rows of the truth table are those where all of

the premises are true– only one premise being false makes the above conjunction false– but a false hypothesis for a conditional can never make the

conditional's result be false

The validity of an argument can be determined from the truth value of the conclusion in the critical rows– it's a valid argument if and only if the conclusion is true in all of

the critical rows– it's an invalid argument if the conclusion is false in any of the

critical rows

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Discrete StructuresCMSC 250Lecture 4

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Rules of Inference (Table 1.3.1- Page 39)

Modus Ponens p q

p

q

Modus Tollens p q

~q

~p

Elimination

p q p q~q ~p

p q

Generalization

p q

p q p q

Specialization

p ^ q p ^ q

p q

Contradiction rule ~p c

p

Transitivity p q

q r

p r

Conjunction

p

q

p ^ q

Proof by division into cases

p qp r

q r

r

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Proofs using rules of inference

P1 p q

P2 ~(q r)P3 p (m r)

~m

P1 p ^ q

P2 p s

P3 ~r ~q

s ^ r

P1 p qP2 q r

P3 ~p

r

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Conditional worlds

Making assumptions are only allowed if you go into a "conditional world"

Two types of results can be derived when an assumption is made: an implication and a negation

list of statements that are true in all worldsassume anything

a list of statements which are true in the situations where the assumption is true

conclusion: the assumption anything from the conditional world

We'll call this "conditional world without contradiction"

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Conditional world assumption leads to contradiction

If you make an assumption, and that assumption leads to a contradiction in the conditional world, what can be concluded?

list of statements that are true in all worlds assume anything

a list of statements which are true in the situations where the assumption is true

a contradiction with something else known to be true

conclusion: the assumption must be false in all possible worlds

We'll call this "conditional world with contradiction"

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Prove using the conditional world methods

P1 (p q) s

P2 r p

r s

P1 m s

P2 s (q ^ r)

P3 q ~r

~(m ^ p)

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Discrete StructuresCMSC 250Lecture 5

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Use both conditional world methods

P1 ~ m pP2 p (q s)

P3 ~(s ~x)

P4 q ~r

~(m ^ r)