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Conditional statements
Conditional statements
If an angle is a right angle, then it
has a measure of 90º.
Conditional statements
If an angle is a right angle, then it
has a measure of 90º.
~p ("not p") is the negation of p.
Conditional statements
If an angle is a right angle, then it
has a measure of 90º.
~p ("not p") is the negation of p.
~p is "An angle is not a right
angle."
Conditional statements
If an angle is a right angle, then it
has a measure of 90º.
~p ("not p") is the negation of p.
~p is "An angle is not a right
angle."
~q is "An angle does not have a
measure of 90º."
Statement:
p
q
Statement:
p
Converse:
q
q
p
Statement:
p
q
Converse:
q
p
Inverse:
~p
~q
Statement:
p
q
Converse:
q
p
Inverse:
~p
~q
Contrapositive:
~q
~p
Statement:
p
q
Converse:
q
p
If an angle is a right angle,
then it has a measure of 90º.
Inverse:
~p
Contrapositive:
~q
~q
~p
Statement:
p
q
Converse:
q
p
If an angle is a right angle,
then it has a measure of 90º.
If an angle has a measure of 90º,
then it is a right angle.
Inverse:
~p
Contrapositive:
~q
~q
~p
Statement:
p
q
Converse:
q
p
If an angle is a right angle,
then it has a measure of 90º.
If an angle has a measure of 90º,
then it is a right angle.
Inverse:
~p
~q
If an angle is a not right angle,
then it does not have a measure of 90º.
Contrapositive:
~q
~p
Statement:
p
q
Converse:
q
p
If an angle is a right angle,
then it has a measure of 90º.
If an angle has a measure of 90º,
then it is a right angle.
Inverse:
~p
~q
If an angle is a not right angle,
then it does not have a measure of 90º.
Contrapositive:
~q
~p
If an angle does not have a measure of 90º,
then it is not a right angle.
Statement:
p
q
Converse:
q
p
If an angle is a right angle,
then it has a measure of 90º.
true
If an angle has a measure of 90º,
then it is a right angle.
Inverse:
~p
~q
If an angle is a not right angle,
then it does not have a measure of 90º.
Contrapositive:
~q
~p
If an angle does not have a measure of 90º,
then it is not a right angle.
Statement:
p
q
true
Converse:
q
p
true
If an angle is a right angle,
then it has a measure of 90º.
If an angle has a measure of 90º,
then it is a right angle.
Inverse:
~p
~q
If an angle is a not right angle,
then it does not have a measure of 90º.
Contrapositive:
~q
~p
If an angle does not have a measure of 90º,
then it is not a right angle.
Statement:
p
q
true
Converse:
q
p
true
If an angle is a right angle,
then it has a measure of 90º.
If an angle has a measure of 90º,
then it is a right angle.
Inverse:
~p
~q
true
If an angle is a not right angle,
then it does not have a measure of 90º.
Contrapositive:
~q
~p
If an angle does not have a measure of 90º,
then it is not a right angle.
Statement:
p
q
true
Converse:
q
p
true
If an angle is a right angle,
then it has a measure of 90º.
If an angle has a measure of 90º,
then it is a right angle.
Inverse:
~p
~q
true
If an angle is a not right angle,
then it does not have a measure of 90º.
Contrapositive:
~q
~p
true
If an angle does not have a measure of 90º,
then it is not a right angle.
Statement:
p
q
Converse:
q
p
If two angles are right angles,
then they are congruent.
Inverse:
~p
Contrapositive:
~q
~q
~p
true
Statement:
p
q
true
Converse:
q
p
false
If two angles are right angles,
then they are congruent.
If two angles are congurent,
then they are right angles.
Inverse:
~p
Contrapositive:
~q
~q
~p
Statement:
p
q
true
Converse:
q
p
false
~q
false
If two angles are right angles,
then they are congruent.
If two angles are congurent,
then they are right angles.
Inverse:
~p
If two angles are not right angles,
then they are not congruent.
Contrapositive:
~q
~p
Statement:
p
q
true
Converse:
q
p
false
~q
false
If two angles are right angles,
then they are congruent.
If two angles are congurent,
then they are right angles.
Inverse:
~p
If two angles are not right angles,
then they are not congruent.
Contrapositive:
~q
~p
true
If two angles are not congurent,
then they are not (both) right angles.
Statement:
p
q
If you live in Chicago,
then you live in Illinois.
Converse:
Inverse:
q
~p
Contrapositive:
p
~q
~q
~p
true
Statement:
p
q
If you live in Chicago,
then you live in Illinois.
Converse:
q
p
If you live in Illinois,
then you live in Chicago.
Inverse:
~p
Contrapositive:
~q
~q
~p
true
false
Statement:
p
q
If you live in Chicago,
then you live in Illinois.
Converse:
q
p
If you live in Illinois,
then you live in Chicago.
Inverse:
~p
~q
true
false
false
If you do not live in Chicago,
then you do not live in Illinois.
Contrapositive:
~q
~p
Statement:
p
q
If you live in Chicago,
then you live in Illinois.
Converse:
q
p
If you live in Illinois,
then you live in Chicago.
Inverse:
~p
~q
true
false
false
If you do not live in Chicago,
then you do not live in Illinois.
Contrapositive:
~q
~p
true
If you do not live in Illinois,
then you do not live in Chicago.
Statement:
p
q
Converse:
q
p
If you live in Illinois,
then you live in Chicago.
Inverse:
~p
Contrapositive:
~q
~q
~p
false
Statement:
p
q
false
Converse:
q
p
true
If you live in Illinois,
then you live in Chicago.
If you live in Chicago,
then you live in Illinois.
Inverse:
~p
Contrapositive:
~q
~q
~p
Statement:
p
q
false
Converse:
q
p
true
If you live in Illinois,
then you live in Chicago.
If you live in Chicago,
then you live in Illinois.
Inverse:
~p
~q
If you do not live in Illinois,
then you do not live in Chicago.
Contrapositive:
~q
~p
true
Statement:
p
q
false
Converse:
q
p
true
If you live in Illinois,
then you live in Chicago.
If you live in Chicago,
then you live in Illinois.
Inverse:
~p
~q
If you do not live in Illinois,
then you do not live in Chicago.
Contrapositive:
~q
~p
If you do not live in Chicago,
then you do not live in Illinois.
true
false
Statement:
p
q
Converse:
q
p
If you live in Indiana
then you live in Chicago.
Inverse:
~p
Contrapositive:
~q
~q
~p
false
Statement:
p
q
false
Converse:
q
p
false
If you live in Indiana
then you live in Chicago.
If you live in Chicago,
then you live in Indiana.
Inverse:
~p
Contrapositive:
~q
~q
~p
Statement:
p
q
false
Converse:
q
p
false
If you live in Indiana
then you live in Chicago.
If you live in Chicago,
then you live in Indiana.
Inverse:
~p
~q
If you do not live in Indiana
then you do not live in Chicago.
Contrapositive:
~q
~p
false
Statement:
p
q
false
Converse:
q
p
false
If you live in Indiana
then you live in Chicago.
If you live in Chicago,
then you live in Indiana.
Inverse:
~p
~q
If you do not live in Indiana
then you do not live in Chicago.
Contrapositive:
~q
~p
If you do not live in Chicago,
then you do not live in Indiana.
false
false
Theorem 3: If a conditional
statement is true, then the
contrapositive of the statement is
also true.
(If p, then q
If ~p, then ~q)
Theorem 3: If a conditional
statement is true, then the
contrapositive of the statement is
also true.
(If p, then q
If ~p, then ~q)
means "if and only if"
Theorem 3: If a conditional
statement is true, then the
contrapositive of the statement is
also true.
(If p, then q
If ~p, then ~q)
means "if and only if"
"iff" means "if and only if"
Circle diagrams
Venn diagrams
Euler diagrams
Circle diagrams
Venn diagrams
Euler diagrams
A
Circle A represents
people who live in
Chicago.
Circle diagrams
Venn diagrams
Euler diagrams
B
A
Circle A represents
people who live in
Chicago.
Circle B represents
people who live in
Illinois.
Circle diagrams
Venn diagrams
Euler diagrams
B
A
Circle A represents
people who live in
Chicago.
Circle B represents
people who live in
Illinois.
Mayor of Chicago
Circle diagrams
Venn diagrams
Euler diagrams
Narrator
B
A
Circle A represents
people who live in
Chicago.
Circle B represents
people who live in
Illinois.
Mayor of Chicago
Circle A represents
people who live in
Chicago.
A
Circle A represents
people who live in
Chicago.
B
A
Circle B represents
people who live in
Illinois.
Circle A represents
people who live in
Chicago.
B
A
C
Circle B represents
people who live in
Illinois.
Circle C represents
people who live in
the USA.
Circle A represents
people who live in
Chicago.
Circle C represents
people who live in
the USA.
B
A
C
Circle B represents
people who live in
Illinois.
D
Circle D represents
people who live in
North America.
Chains of Reasoning
Chains of Reasoning
If p
q, and q
then p r.
r,
Chains of Reasoning
If p
q, and q
then p r.
r,
If you live in Chicago, then you live in Illinois.
Chains of Reasoning
If p
q, and q
then p r.
r,
If you live in Chicago, then you live in Illinois.
If you live in Illinois, then you live in the USA.
Chains of Reasoning
If p
q, and q
then p r.
r,
If you live in Chicago, then you live in Illinois.
If you live in Illinois, then you live in the USA.
Conclusion: If you live in Chicago, then you live
in the USA.
Chains of Reasoning
If p
q, q
r, and r
then p s.
s,
Chains of Reasoning
If p
q, q
r, and r
then p s.
s,
If you live in Chicago, then you live in Illinois.
Chains of Reasoning
If p
q, q
r, and r
then p s.
s,
If you live in Chicago, then you live in Illinois.
If you live in Illinois, then you live in the USA.
Chains of Reasoning
If p
q, q
r, and r
then p s.
s,
If you live in Chicago, then you live in Illinois.
If you live in Illinois, then you live in the USA.
If you live in the USA then you live in North
America.
Chains of Reasoning
If p
q, q
r, and r
then p s.
s,
If you live in Chicago, then you live in Illinois.
If you live in Illinois, then you live in the USA.
If you live in the USA then you live in North
America.
Conclusion: If you live in Chicago, then you live
in North America.
If I study for many hours, I will do well in
geometry.
If I study for many hours, I will do well in
geometry.
If I do well in geometry, I will get into a good
college.
If I study for many hours, I will do well in
geometry.
If I do well in geometry, I will get into a good
college.
If I get into a good college, I will be able get a
high paying job.
If I study for many hours, I will do well in
geometry.
If I do well in geometry, I will get into a good
college.
If I get into a good college, I will be able get a
high paying job.
If I get a high paying job, I will become wealthy.
If I study for many hours, I will do well in
geometry.
If I do well in geometry, I will get into a good
college.
If I get into a good college, I will be able get a
high paying job.
If I get a high paying job, I will become wealthy.
Conclusion: If I study for many hours, I will
become wealthy.
Conditional statement:
p
q
Conditional statement:
Converse:
p
q
q
p
Conditional statement:
Converse:
Inverse:
p
q
~p
q
p
~q
Conditional statement: p
Converse: q
Inverse: ~p
Contrapositive: ~q
q
p
~q
~p
Conditional statement: p
Converse: q
Inverse: ~p
Contrapositive: ~q
q
p
~q
~p
Circle diagrams (also know as Venn
or Euler diagrams) help us depict
conditional statements.
Conditional statement: p
Converse: q
Inverse: ~p
Contrapositive: ~q
q
p
~q
~p
Circle diagrams (also know as Venn
or Euler diagrams) help us depict
conditional statements.
means "if and only if."
Theorem 3: If a conditional
statement is true, then the
contrapositive of the statement is
also true.
(If p, then q
If ~p, then ~q)