Discrete Structures
Chapter 3: The Logic of Quantified Statements
3.2 Predicates and Quantified Statements II
TOUCHSTONE: Stand you both forth now: stroke your chins, and
swear by your beards that I am a knave.
CELIA: By our beards – if we had them – thou art.
TOUCHSTONE: By my knavery – if I had it – then I were; but if
you swear by that that is not, you are not forsworn.
– William Shakespeare, 1564 – 1616
As You Like It, 1600
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Theorem
• Negation of a Universal Statement
– The negation of a statement of the form
 x in D, Q(x)
is logically equivalent to a statement of the form
 x in D such that Q(x).
Symbolically, ( x  D, Q(x))   x  D s.t.
Q(x).
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Example
• Consider the statement:
“All scientists wear glasses.”
Possible Negation: “No scientist wear
glasses.”
Correct Negation: “There is at least one
scientist who does not wear glasses.”
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And this means…
The negation of a universal statement (“all
are”) is logically equivalent to an existential
statement (“some are not” or “there is at least
one that is not”).
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Theorem
• Negation of an Existential Statement
– The negation of a statement of the form
 x in D such that Q(x)
is logically equivalent to a statement of the form
 x in D,  Q(x).
Symbolically, ( x  D, s.t. Q(x))   x  D,
Q(x).
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Example
• Consider the statement:
“Some snowflakes are the same.”
Negation: “No snowflakes are the same.”
“All snowflakes are different.”
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And this means…
The negation of an existential statement
(“some are”) is logically equivalent to a
universal statement (“none are” or “all are
not”).
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Example – pg. 116 # 2
• Which of the following is a negation for “All dogs
are loyal”? More that one answer may be correct.
a.
b.
c.
d.
e.
f.
g.
h.
All dogs are disloyal.
No dogs are loyal.
Some dogs are disloyal.
Some dogs are loyal.
There is a disloyal animal that is not a dog.
There is a dog that is disloyal.
No animals that are not dogs are loyal.
Some animals that are not dogs are loyal.
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Example – pg. 116 #3
• Write a formal negation for each of the
following statements:
–  fish x, x has gills.
–  computers c, c has a CPU.
–  a movie m such that m is over 6 hours long.
–  a band b such that b has won at least 10 Grammy
awards.
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Negation of a Universal
Conditional Statement
•
~   x, if P  x  then Q  x     x such that P  x  and ~ Q  x  .
• Written more symbolically
~   x, P  x   Q  x     x s.t.  P  x   ~ Q  x   .
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Example – pg. 116 # 17
• Write a negation for the statement below.
–  integers d, if 6/d is an integer then d = 3.
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Variants of Universal Conditional
Statements
• Remember (2.2) that a conditional statement
has a contrapositive, a converse, and an
inverse. The definitions can be extended to
universal conditional statements as shown on
the next page.
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Definition
• Consider a statement of the form:
x  D, if P( x) then Q( x).
1. Its contrapositive is the statement:
x  D, if ~Q( x) then ~P( x).
2. Its converse is the statement:
x  D, if Q( x) then P( x).
3. Its inverse is the statement:
x  D, if ~P( x) then ~Q( x).
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Example – pg. 117 # 27
• For each statement write the converse, inverse,
and contrapositive. Indicate as best as you can
which among the statement, its converse, its
inverse, and its contrapositive are true and
which are false. Give a counterexample for
each that is false.
–  integers d, if 6/d is an integer then d = 3.
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Necessary and Sufficient
Conditions, Only If
• Definition
– The definitions of necessary, sufficient, and only if can also
be extended to apply to universal conditional statements.
1. "x, r  x  is a sufficient condition for s  x " means "x, if r  x  then s  x ".
2. " x, r  x  is a neccesary condition for s  x  " means
"x, if ~r  x  then ~s  x  " or equivalently, "x, if s  x  then r  x  "
3. " x, r  x  only if s  x  " means "x, if ~s  x  then ~r  x  " or,
equivalently, "x, if r  x  then s  x  ".
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Example – pg. 117 #44 & 46
• Use the facts that the negation of a  statement is a 
statement and that the negation of an if-then statement
is an and statement to rewrite the statements without
using the word necessary or sufficient.
– Having a large income is not a necessary condition for a
person to be happy.
– Being a polynomial is not a sufficient condition for a
function to have a real root.
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