2. Propositional Equivalences 2.1. Tautology/Contradiction
Build a truth table to verify that the proposition (p ↔ q)∧(¬p∧q) is a contradiction. 2.2. Logically Equivalent. Definition 2.2.1. Propositions r and s are
Propositional Logic Discrete Mathematics
A compound proposition that is always True is called a tautology. Two propositions p and q are logically equivalent if their truth tables are the same. Namely
CHAPTER 2 1. Logic Definitions 1.1. Propositions. Definition 1.1.1. A
When building a truth table for a compound proposition you need a row for every Here we build a truth table for p → (q → r) and (p ∧ q) → r. When ...
Chapter 3 Review Finite Math Name: ANSWER KEY
Oct 16 2014 Let p represent a true statement
Compound Propositions 1
Make the truth table in the same way as for the conjunction and disjunction. A conditional is false only when T→F. p q p → q. 1. T.
THE CONDITIONAL TRUTH TABLE FOR THE CONDITIONAL
From this we see that compound propositions P and Q are logically equivalent when and only when P Q is a tautology. THE BICONDITIONAL. Note. P Q is
MA0301 ELEMENTARY DISCRETE MATHEMATICS NTNU
Jan 6 2020 What can you say about the two compound propositions p ∨ (q ∧ r) and ... Write down the truth table for (p ⇒ q) ⇒ (p ∧ q). Solution. The ...
Module 1: Basic Logic Theme 1: Propositions
The reader may inspect the truth table of p →q in Table 4 below. Table 4 We say that two compound propositions P and Q are logically equivalent if they have ...
Propositional Logic
Sep 19 2008 atomic proposition p; and the proposition (q → s) is made up of the ... The truth values for ¬p are obtained using the truth table for negation ...
2. Propositional Equivalences 2.1. Tautology/Contradiction
Show that (p ? q) ? (q ? p) is logically equivalent to p ? q. Solution 1. Show the truth values of both propositions are identical. Truth Table: p q p ?
Propositional Logic
Truth tables for compound propositions. Construct the truth table for the compound proposition: (p ? ¬q) ? (p ? q). p q. ¬q p ? ¬q p ? q (p ? ¬q)
Propositional Logic Discrete Mathematics
The truth table lists all possible combinations of the values of the Look at the following two compound propositions: p ? q and q ? ¬p. p q p ? q.
Chapter 3 Review Finite Math Name: ANSWER KEY
Let p represent a true statement while q and r represent false statements. Find the truth value of the compound statement. 12) ?[(?p ? ?q) ? ?q].
Propositional Logic
Truth tables for compound propositions. Construct the truth table for the compound proposition: (p ? ¬q) ? (p ? q). p q. ¬q p ? ¬q p ? q (p ? ¬q)
CHAPTER 2 1. Logic Definitions 1.1. Propositions. Definition 1.1.1. A
p ? q: This book is interesting and I am staying at home. Truth Table: p q p ? q. T T. T. T F. F. F T. F.
Propositional Logic
19 ??? 2008 Basic Terms: atomic propositions represented by the symbols: p q
Math 55: Discrete Mathematics
h) ¬q ? (¬p ? q) : The votes have not been counted or they have 1.1.38 Construct a truth table for ((p ? q) ? r) ? s. p q p ? q r. (p ? q) ? r.
Propositional Logic
Truth tables for compound propositions. Construct the truth table for the compound proposition: (p ? ¬q) ? (p ? q). p q. ¬q p ? ¬q p ? q (p ? ¬q)
Chapter 1 Part I: Propositional Logic
ó Compound Propositions; constructed from logical Compound Propositions: Negation ... ó Construct a truth table for p q r. ¬r p ? q p ? q ? ¬r.
Propositional Logic BasicsPropositional EquivalencesNormal formsBoolean functions and digital circuits
Propositional Logic
Lucia Moura
Winter 2012
CSI2101 Discrete Structures Winter 2012: Propositional LogicLucia Moura
Propositional Logic BasicsPropositional EquivalencesNormal formsBoolean functions and digital circuits
Propositional Logic: Section 1.1
Proposition
A proposition is a declarative sentence that is either true or false. Which ones of the following sentences are propositions?Ottawa is the capital of Canada.Buenos Aires is the capital of Brazil.
2 + 2 = 4
2 + 2 = 5
if it rains, we don't need to bring an umbrella. x+ 2 = 4x+y=zWhen does the bus come?Do the right thing.
CSI2101 Discrete Structures Winter 2012: Propositional LogicLucia Moura
Propositional Logic BasicsPropositional EquivalencesNormal formsBoolean functions and digital circuits
Propositional Logic: Section 1.1
Propositional variable and connectives
We use lettersp;q;r;:::to denotepropositional variables(variables that represent propositions). We can form new propositions from existing propositions usinglogical operatorsorconnectives. These new propositions are calledcompound propositions.Summary of connectives:namenicknamesymbol
negationNOT: conjunctionAND^ disjunctionOR_ exclusive-ORXOR implicationimplies! biconditionalif and only if$ CSI2101 Discrete Structures Winter 2012: Propositional LogicLucia Moura
Propositional Logic BasicsPropositional EquivalencesNormal formsBoolean functions and digital circuits
Propositional Logic: Section 1.1
Meaning of connectivesp q:pp^qp_qpqp!qp$qT TFTTFTTT FFFTTFF
F TTFTTTF
F FTFFFTT
WARNING:
Implication (p!q) causes confusion, specially in line 3: \F!T" is true. One way to remember is that the rule to be obeyed is \if the premisepis true then the consequenceqmust be true."The only truth assignment that falsies this isp=Tandq=F.CSI2101 Discrete Structures Winter 2012: Propositional LogicLucia Moura
Propositional Logic BasicsPropositional EquivalencesNormal formsBoolean functions and digital circuits
Propositional Logic: Section 1.1
Truth tables for compound propositions
Construct the truth table for the compound proposition: (p_ :q)!(p^q)p q:qp_ :qp^q(p_ :q)!(p^q)T TF T FT F TF F FT CSI2101 Discrete Structures Winter 2012: Propositional LogicLucia Moura
Propositional Logic BasicsPropositional EquivalencesNormal formsBoolean functions and digital circuits
Propositional Equivalences: Section 1.2
Propositional Equivalences
A basic step is math is to replace a statement with another with the same truth value (equivalent). This is also useful in order to reason about sentences.Negate the following phrase:
\Miguel has a cell phone and he has a laptop computer."p="Miguel has a cell phone"q=\Miguel has a laptop computer."The phrase above is written as(p^q).Its negation is:(p^q), which is logically equivalent to:p_ :q.
(De Morgan's law)This negation therefore translates to: \Miguel does not have a cell phone or he does not have a laptop computer." CSI2101 Discrete Structures Winter 2012: Propositional LogicLucia Moura
Propositional Logic BasicsPropositional EquivalencesNormal formsBoolean functions and digital circuits
Propositional Equivalences: Section 1.2
Truth assignments, tautologies and satisabilityDenitionLetXbe a set of propositions.
Atruth assignment(toX) is a function:X! ftrue;falsegthat assigns to each propositional variable a truth value. (A truth assignment corresponds to one row of the truth table) If the truth value of a compound proposition under truth assignmentistrue, we say thatsatisesP, otherwise we say thatfalsiesP.A compound propositionPis atautologyif every truth assignment
satisesP, i.e. all entries of its truth table aretrue.A compound propositionPissatisableif there is a truth assignment
that satisesP; that is, at least one entry of its truth table is true.A compound propositionPisunsatisable (or a contradiction)if it
is not satisable; that is, all entries of its truth table are false. CSI2101 Discrete Structures Winter 2012: Propositional LogicLucia Moura
Propositional Logic BasicsPropositional EquivalencesNormal formsBoolean functions and digital circuits
Propositional Equivalences: Section 1.2
Examples: tautology, satisable, unsatisable
For each of the following compound propositions determine if it is atautology, satisable or unsatisable:(p_q)^ :p^ :qp_q_r_(:p^ :q^ :r)(p!q)$(:p_q)CSI2101 Discrete Structures Winter 2012: Propositional LogicLucia Moura
Propositional Logic BasicsPropositional EquivalencesNormal formsBoolean functions and digital circuits
Propositional Equivalences: Section 1.2
Logical implication and logical equivalenceDenition A compound propositionplogically impliesa compound propositionq (denotedp)q) ifp!qis a tautology. Two compound propositionspandqarelogically equivalent(denoted pq, orp,q) ifp$qis a tautology.Theorem Two compound propositionspandqare logically equivalent if and only ifplogically impliesqandqlogically impliesp.In other words: two compound propositions are logically equivalent if and
only if they have the same truth table. CSI2101 Discrete Structures Winter 2012: Propositional LogicLucia Moura
Propositional Logic BasicsPropositional EquivalencesNormal formsBoolean functions and digital circuits
Propositional Equivalences: Section 1.2
Logically equivalent compound propositions
Using truth tables to prove that(p!q)and:p_qare logically equivalent, i.e. (p!q) :p_qp q:p:p_qp!qT TFTTT FFFF
F TTTT
F FTTT
What is the problem with this approach?
CSI2101 Discrete Structures Winter 2012: Propositional LogicLucia Moura
Propositional Logic BasicsPropositional EquivalencesNormal formsBoolean functions and digital circuits
Propositional Equivalences: Section 1.2
Truth tables versus logical equivalences
Truth tables grow exponentially with the number of propositional variables!A truth table withnvariables has2nrows.
Truth tables are practical for small number of variables, but if you have, say, 7 variables, the truth table would have 128 rows! Instead, we can prove that two compound propositions are logically equivalent by using known logical equivalences (\equivalence laws"). CSI2101 Discrete Structures Winter 2012: Propositional LogicLucia Moura
Propositional Logic BasicsPropositional EquivalencesNormal formsBoolean functions and digital circuits
Propositional Equivalences: Section 1.2
Summary of important logical equivalences I
Note T is the compound composition that is always true, and F is the compound composition that is always false.
CSI2101 Discrete Structures Winter 2012: Propositional LogicLucia Moura
Propositional Logic BasicsPropositional EquivalencesNormal formsBoolean functions and digital circuits
quotesdbs_dbs2.pdfusesText_4[PDF] construction class 9 ncert pdf
[PDF] construction company balance sheet pdf
[PDF] construction engineering and management pdf
[PDF] construction floor area
[PDF] construction management books by indian authors pdf
[PDF] construction management notes for civil engineering pdf
[PDF] construction site management pdf
[PDF] constructor and destructor for struct in c++
[PDF] constructor and destructor in c++ example program
[PDF] constructor and destructor in c++ pdf download
[PDF] constructor and destructor in c ppt
[PDF] constructor can be overloaded or not
[PDF] constructor can be overloaded true or false
[PDF] constructor overriding in java
