Conceptual Foundations of Computing

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Conceptual Foundations of Computing

• Lecture 1

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Statements
A Statement (or proposition) is a sentence that
is true or false but not both.
Compound Statements
Compound statements are made by joining simple
statements together using logical operators, like
not, and and or.
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Negation
If p is a statement variable, the negation of p
is not p or It is not the case that p and is
denoted by p. It has opposite truth value from
p if p is true, p is false if p is false, p s
true.
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Conjunction
If p and q are statement variables, the
conjunction of p and q is p and q, donated by
p ? q. It is true when, and only when, both p and
q are true. If either p or q is false, or if both
are false, p q is false.
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Logic on Excel
Excel has logical functions. The following set of
results for the p and q column where generated
using the table entry in cell C2 AND(A2,B2)
(and duplicating it down the column.
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Disjunction
If p and q are statement variables, the
disjunction of p and q is p or q, denoted by p
V q. It is true when either p is true or q is
true or both p and q are true it is false only
when both p and q are false.
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Evaluating Compound Statements
A statement form (or propositional form) is an
expression made up of statement variables (such
as p, q, r) and logical connectives (such as , ?
and V) that becomes a statement when actual
statements are substituted for the component
statement variables. The Truth table for a given
statement form displays the truth values that
correspond to all possible combinations of truth
values for its component statement variables.
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Example Exclusive Or (XOR)
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Truth Table for (p ? q) V r
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Exercise
Extend pervious truth table Develop a
column for (p ? q) V (p ? r) and a column for p
? (q V r). Notice anything about these two
columns?
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Logical Equivalence
Two statement forms are called logically
equivalent if, and only if, they have identical
truth values for each possible substitution of
statements for their statement variables. The
logical equivalence of statement forms P and Q is
denoted by writing . Two
statements are called logically equivalent if,
and only if, they have logically equivalent forms
when identical component statements variables are
used to replace identical component statements.
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Logical Equivalence Example
The Distributed laws (page 14 of Textbook) tell
us that p ? (q V r) (p ? q) V (p ? r) p V
(q ? r) (p V q) ? (p V r) And, consequently
(substituting r for r) p ? (q V r) (p ?
q) V (p ? r)
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Double negative Property
Two negatives make a positive I was not
absent (-1) (-1) 1
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Showing Nonequivalence
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De Morgans Laws
The negation of an and statement is logically the
equivalent to the or statement in which each
component is negated. (p ? q) p V q The
negation of an or statements is logically
equivalent to the and statement in which each
component is negated. (p V q) p ? q
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One of De Morgans Laws
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Tautologies and Contradictions
A tautology is a statement form that is always
true regardless of the truth values of the
individual statements substituted for its
statement variables. A statement whose form is a
tautology is a tautological statement. A
contradiction is a statement form that is always
false regardless of the truth values of the
individual statements substituted for its
statement variables. A statements whose form is a
contradiction is a contradictory statement.
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Logical Equivalence Involving T and C
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Simplifying Statement Forms
This is logics equivalent of simplifying maths
problems x 6(y 5) 2
(6y 30) 2 6y 32
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Conditional Statements
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Introduction
When you make a logical inference or deduction,
you reason from a hypothesis to a conclusion.
Your aim is to be able to say, If such and
such is known, then something or other must be
the case.
If 4,686 is divisible by 6 then, 4,686 is
divisible by 3.
We can do this because we Know that 6 2 3.
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Example

• If you show up for work on Monday, then you will
  get the job.
• When is this statement false?
• When do you get the job?
• What happens if you do not turn up? Is the
  statement true or false if you do not show?

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Conditional Statements
If p and q are statement variables, the
conditional of p by q is if p then q or p
implies q and is donated by p ? q. It is false
when p is true and q is false otherwise it is
true. We call p the hypothesis (or antecedent) of
the conditional and q the conclusion (or
consequent).
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Conditional Statements
From page 18 of the Textbook A conditional
statement that is true by virtue of the fact that
its hypothesis is false is often called vacuously
true or true by default. Thus the statement If
you show up for work on Monday morning, then you
will get the job is vacuously true if you do not
show up for work on Monday.
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Logical Equivalences involving ?
Suppose you know the truth of r follows from the
truth of p and also follows from the truth of q.
Then no matter whether p or q is the case, the
truth of r must follow. The division into
cases method of analysis (simplification?) is
based upon this idea.
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IF-Then as Or
If then statements can be rewritten in terms of
not and or operators
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Negation of a Conditional Statement
The negation of If p then q is logically
equivalent to p and not q.
Also working symbolically. We start with
and negate both sides
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The Contrapositive of a CS
The contrapositive of a conditional statement of
the form If p then q is If q then
p Symbolically, the contrapositive of p ?
q is q ? p A conditional statement is
logically equivalent to its contrapositive p
? q q ? p
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Converse and Inverse
Suppose a conditional statement of the form If p
then q is given. The converse of this statement
is If q then p. The inverse of this statement
is If p then q. Symbolically, the converse
of p ? q is q ? p the inverse of
p ? q is p ? q Exercise Are
these statements logically equivalent?
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Converse and Inverse
contrapositive p ? q q ?
p converse p ? q is not logically
equivalent to q ? p inverse p ? q
is not logically equivalent to p ? q It
is a common mistake to read a conditional
statement as its converse. To pass this course
you must pass the exam. p pass the course
q pass the exam p ? q If you pass the
course then you must (have) pass(ed) the exam. A
student may mistakenly take the statement to
mean q ? p If you pass the exam then you pass
the course. However, there may be other
conditions that must be met to pass the course.
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Only If and the Biconditional
p if q means p is True if q is True if q
then p q ? p p only if q means p is True
only if q is True if q then p q ? p p ?
q (contrapositive) p if and only if q
means p if q and p only if q (q ? p)
(p ? q) also known as biconditional of p and
q p ? q
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Biconditional, Converse and Inverse
A conditional statement with form p ? q, like
you pass the course only if you pass the exam,
may mistakenly read as the biconditional
statement p ? q, you pass the course if and only
if you pass the exam. The biconditional p if
an only if q is (q ? p) (p ? q). If this
statement is True, then the following statements
are all True p ? q q ? p (converse of p ?
q) p ? q (inverse of p ? q contrapositive of
q ? p) Reading a conditional statement as a
biconditional can result in the statement being
represented by its converse. Note We will
study methods of valid argument (deducing new
True statements from a set of given True
statements) next week.
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Precedence of Logical Operators

• The five logical operators have the following
  precedence (order of application). Remember to
  use brackets() to change this order.
• Evaluate negation first
• ?, V Evaluate and and or second Use
  brackets when both are present.
• ?, ? Evaluate conditional and
  biconditional third. Use brackets when both
  are present.

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Necessary and Sufficient Conditions
In mathematics and computing we often talk about
necessary and sufficient conditions. p is a
sufficient condition for q means If p then
q. p is a necessary condition for q means if
not p then not q. p ? q q ? p
(contrapositive) if q then p passing
the exam is a necessary condition for passing the
course to pass the course you must pass the
exam p is a necessary and sufficient
condition for q means if p then q and if q
then p this is the biconditional of p and q
p ? q
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Something to think about
p It is Wednesday. q It is raining. If p is
False, what is the value of p ? q ? p ? q is
only False when p is True and q is False. So, if
p is False, p ? q is True. So, It is Wednesday
implies it is raining. is True? Do you feel
comfortable about that? More on this next week.