MGF 1106 – Section 3.4a – Negations
A. REVIEW:
To negate a simple statement insert (or delete) "not"
Given:
Chad is an LSCC student.
Negation: Chad is not an LSCC student.
To negate a statement with a quantifier use the "box"
All are
None are
Given:
Some movies are comedies.
Negation: No movies are comedies.
Some are
Some are not
To negate a compound statement, use it is false that:
Given:
The sky is blue and the clouds are white.
Negation: It is false that, the sky is blue and the clouds are white.
NOTE: When negating a negation you get a positive statement.
Ex: ~ (~ p) is p
Also, recall logically equivalent means that statements have the same truth value (or their answer
columns are exactly the same.
B. NEXT STEP:
DeMorgan’s Laws: 1) ~ (p ^ q)  ~ p v ~q
2) ~ (p v q)  ~ p ^~ q
Application: Use DeMorgan's Laws (rather than truth tables) to determine if two statements
are logically equivalent.
1. ~ (~p v q)
p  q
2. ~p v q
~ (p  ~q)
Complete the following by finding a logically equivalent statement using DeMorgan’s laws:
Negate the following symbolic statements.
1. ~ (~p v q) 
2. ~ (p ^ ~q) 
3. ~ (~p ^ q) 
4. ~ (p v ~ q) 
To change a conditional statement to a disjunction, look at these truth tables:
p
T
q
T
T
F
F
F
T
F
p q
p
T
q
T
T
F
F
F
T
F
~p v q
So, to change a conditional statement to a logically equivalent disjunction, negate the p term,
change the statement to a disjunction, and keep the q term the same.
1. p  ~q

2. ~p  q

3. p  (q v r) 
.
MGF 1106 – Section 3.4 b – Equivalent Statements - Conditionals
A. Forms of a Conditional
Given the statement:
if p then q
The Converse is:
if q then p
The Inverse is:
if not p then not q
The Contrapositive is:
if not q then not p
p q
q p
~p  ~q
~q  ~p
Example:
Given the statement: If a number ends in zero, then it is divisible by five.
The converse is:
If a number is divisible by five, then it ends in zero.
The inverse is:
If a number does not end in zero, then it is not divisible by five.
The contrapositive is: If a number is not divisible by five ,then it does not end in zero.
You Try: Given the statement, give the converse, inverse and contrapositive.
1. If I hear thunder, then it will rain.
Converse:
Inverse:
Contrapositive:
**2. If the seas are not rough, then some boats will sail.
Converse:
Inverse:
Contrapositive:
B. Equivalences – Use truth tables for the various forms of a conditional to determine which forms
have the same truth values.
Statement
p
T
q
T
T
F
F
F
T
F
Converse
p q
Inverse
p
T
q
T
T
F
F
F
T
F
p
T
q
T
T
F
F
F
T
F
q p
Contrapositive
~p  ~q
Summary: Which forms are EQUIVALENT?
p
T
q
T
T
F
F
F
T
F
~q  ~p
C. Summary: When asked to give "Logically Equivalent" Statements Use one of the following
options:
For conditionals, use the contrapositive:
p  q is equivalent with ~q  ~p
For negations (It is false that), use DeMorgan's Laws:
~ (p ^ q) is equivalent with ~p v ~q
~ (p v q) is equivalent with ~p ^ ~q
For negations with quantifiers use “ the chart.”
Note: When asked to find a statements that are NOT logically equivalent to a conditional,
look for the converse or inverse.
Practice: Give a statement that is logically equivalent to each of the following.
1. If I practice every day then I hit a home run.
2. It is false that all hits are home runs.
3. It is false that I hit a home run or I stole first base.
4. It is false that no snow falls in Florida.
More practice:
5. Give the negation of: I get paid or I will quit.
6. Give the negation of: All dogs bark.
7. Give a statement that is logically equivalent to: If all dogs bark then some neighbors wake up.
8. Give the negation of: If all dogs bark then some neighbors wake up.
9. Give two statements NOT logically equivalent to the statement in problem 7.
D. Use these rules to select the logically equivalent statement:
1. Select the statement below that is logically equivalent to "If the animal is a bird, then it flies."
a. If the animal is not a bird, then it does not fly.
b. If it doesn't fly, then the animal is not a bird.
c. The animal is a bird and it flies.
d. If it flies, then the animal is a bird.
2. Select the statement below that is logically equivalent to "It is not true that some women are astronauts."
a. No woman is an astronaut.
b. If the person is an astronaut, then that person is a woman.
c. All women are astronauts.
d. Some women are not astronauts.
Determine if these 2 statements are logically equivalent. (we can use truth tables)
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