Logic (Geometry)
See how to find the truth value of a statement
by using logic.
30 examples and their solutions.
Statement
Definition
Statement: ( o ) or ( x )
A statement is a sentencethat is either true ( o ) or false ( x ).
Negation
Definition
~p
[~p] is the negation of p.It means [not p].
Example
Find the negation of the given statement.
5 is a positive number.
Solution 5 is a positive number.
p: 5 is a positive number.
~p: 5 is not a positive number.
~p: 5 is not a positive number.
Close
Example
Find the negation of the given statement.
1 + 2 = 0
Solution 1 + 2 = 0
p: 1 + 2 = 0
~p: 1 + 2 ≠ 0
~p: 1 + 2 ≠ 0
Close
Example
Find the negation of the given statement.
2 is not an odd number.
Solution 2 is not an odd number.
p: 2 is not an odd number.
~p: 2 is an odd number. - [1]
~p: 2 is an odd number. - [1]
[1]
Negation of 'is not' → 'is'
Close
Truth Value
p | ~p |
---|---|
o | x |
x | o |
Example
p: 5 is a positive number.
Truth value of ~p?
Solution Truth value of ~p?
p: ( o )- [1]
~p: ( x )
False
~p: ( x )
False
[1]
5 is a positive number.
So p is true.
So p is true.
Close
Example
q: 3 is an even number.
Truth value of ~q?
Solution Truth value of ~q?
q: ( x )- [1]
~q: ( o )
True
~q: ( o )
True
[1]
3 is not an even number.
So q is false.
So q is false.
Close
Example
Solution
Close
r: ( o )- [1]
~r: ( x )
~(~r): ( o )- [2]
True
~r: ( x )
~(~r): ( o )- [2]
True
[1]
[2]
~r is false.
So its negation, ~(~r), is true.
So its negation, ~(~r), is true.
Close
Conjunction
Definition
p ∧ q
[p ∧ q] is the conjunction of p and q.It means [p and q].
Truth Value
p | q | p ∧ q |
---|---|---|
o | o | o |
o | x | x |
x | o | x |
x | x | x |
if both p and q are true.
Example
p: 5 is a positive number.
q: 1 + 1 = 3
Truth value of p ∧ q?
Solution q: 1 + 1 = 3
Truth value of p ∧ q?
p: ( o )
q: ( x )
p ∧ q: ( x )
False
q: ( x )
p ∧ q: ( x )
False
Close
Example
p: 5 is a positive number.
r: 4 > 2
Truth value of p ∧ r?
Solution r: 4 > 2
Truth value of p ∧ r?
p: ( o )
r: ( o )
p ∧ r: ( o )
True
r: ( o )
p ∧ r: ( o )
True
Close
Disjunction
Definition
p ∨ q
[p ∨ q] is the disjunction of p and q.It means [p or q].
Truth Value
p | q | p ∨ q |
---|---|---|
o | o | o |
o | x | o |
x | o | o |
x | x | x |
if either p and q are true.
Example
p: 5 is a positive number.
q: 1 + 1 = 3
Truth value of p ∨ q?
Solution q: 1 + 1 = 3
Truth value of p ∨ q?
p: ( o )
p ∧ q: ( o )
True
p ∧ q: ( o )
True
p is true.
Then p ∧ q is true.
(You don't need to find the truth value of q.)
Then p ∧ q is true.
(You don't need to find the truth value of q.)
Close
Example
q: 1 + 1 = 3
s: 2 > 9
Truth value of q ∨ s?
Solution s: 2 > 9
Truth value of q ∨ s?
q: ( x )
s: ( x )
q ∨ s: ( x )
False
s: ( x )
q ∨ s: ( x )
False
Close
Conditional
Definition
p → q
[p → q] is a conditional statement.It means [if p, then q].
p: Hypothesis
q: Conclusion
Example
Find the hypothesis and conclusion of the given statement.
If 2 is a prime number,
then 2 is an odd number.
Solution If 2 is a prime number,
then 2 is an odd number.
p
If 2 is a prime number,
then 2 is an odd number.
q
Hypothesis: 2 is a prime number.
Conclusion: 2 is an odd number.
If 2 is a prime number,
then 2 is an odd number.
q
Hypothesis: 2 is a prime number.
Conclusion: 2 is an odd number.
Close
Example
Find the hypothesis and conclusion of the given statement.
If he is not in his room,
then he is playing basketball.
Solution If he is not in his room,
then he is playing basketball.
p
If he is not in his room,
then he is playing basketball.
q
Hypothesis: He is not in his room.
Conclusion: He is playing basketball.
If he is not in his room,
then he is playing basketball.
q
Hypothesis: He is not in his room.
Conclusion: He is playing basketball.
Close
Example
Find the hypothesis and conclusion of the given statement.
I'm staying home if it's raining.
Solution I'm staying home if it's raining.
qp
I'm staying home if it's raining.
Hypothesis: It's raining.
Conclusion: I'm staying home.
I'm staying home if it's raining.
Hypothesis: It's raining.
Conclusion: I'm staying home.
Close
Truth Value
p | q | p → q |
---|---|---|
o | o | o |
o | x | x |
x | o | o |
x | x | o |
Example
p: 2 is a prime number.
q: 2 is a positive number.
Truth value of p → q?
Solution q: 2 is a positive number.
Truth value of p → q?
p: ( o )
q: ( o )
p → q: ( o )
True
q: ( o )
p → q: ( o )
True
Close
Example
p: 2 is a prime number.
r: 2 is an odd number.
Truth value of p → r?
Solution r: 2 is an odd number.
Truth value of p → r?
p: ( o )
r: ( x )
p → r: ( x )
False
r: ( x )
p → r: ( x )
False
Close
Example
p: 2 is a prime number.
r: 2 is an odd number.
Truth value of r → p?
Solution r: 2 is an odd number.
Truth value of r → p?
r: ( x )
r → p: ( o )
True
r → p: ( o )
True
The hypothesis r is false.
Then r → p is true.
(You don't need to find the truth value of p.)
Then r → p is true.
(You don't need to find the truth value of p.)
Close
Inverse
Definition
Example
Solution
Close
p
If 2 is a prime number,
then 2 is an odd number.
q
~p → ~q: If 2 is not a prime number,
then 2 is not an odd number.
If 2 is a prime number,
then 2 is an odd number.
q
~p → ~q: If 2 is not a prime number,
then 2 is not an odd number.
Close
Example
If he is not in his room,
then he is playing basketball.
Inverse?
Solution then he is playing basketball.
Inverse?
p
If he is not in his room,
then he is playing basketball.
q
~p → ~q: If he is in his room,
then he is not playing basketball.
If he is not in his room,
then he is playing basketball.
q
~p → ~q: If he is in his room,
then he is not playing basketball.
Close
Converse
Definition
q → p
To find the converse of [p → q],switch p and q.
Example
Solution
Close
p
If 2 is a prime number,
then 2 is an odd number.
q
q → p: If 2 is an odd number,
then 2 is a prime number.
If 2 is a prime number,
then 2 is an odd number.
q
q → p: If 2 is an odd number,
then 2 is a prime number.
Close
Example
If he is not in his room,
then he is playing basketball.
Converse?
Solution then he is playing basketball.
Converse?
p
If he is not in his room,
then he is playing basketball.
q
q → p: If he is playing basketball,
then he is not in his room.
If he is not in his room,
then he is playing basketball.
q
q → p: If he is playing basketball,
then he is not in his room.
Close
Contrapositive
Definition
Example
Solution
Close
p
If 2 is a prime number,
then 2 is an odd number.
q
~q → ~p: If 2 is not an odd number,
then 2 is not a prime number.
If 2 is a prime number,
then 2 is an odd number.
q
~q → ~p: If 2 is not an odd number,
then 2 is not a prime number.
Close
Example
If he is not in his room,
then he is playing basketball.
Contrapositive?
Solution then he is playing basketball.
Contrapositive?
p
If he is not in his room,
then he is playing basketball.
q
~q → ~p: If he is not playing basketball,
then he is in his room.
If he is not in his room,
then he is playing basketball.
q
~q → ~p: If he is not playing basketball,
then he is in his room.
Close
Law of Contrapositive
Law
p → q = ~q → ~p
A conditional and its contrapositivehave the same truth value.
Relationship between Conditional, Inverse, Converse, and Contrapositive
are the contrapositive of each other.
So, by the law of contrapositive,
the inverse and the converse
also have the same truth value.
Example
If the given statement is true,
write a statement that is always true.
If it's raining, then I'm staying home.
Solution write a statement that is always true.
If it's raining, then I'm staying home.
pq
If it's raining, then I'm staying home.
~q → ~p: If I am not staying home,
then it is not raining.
If it's raining, then I'm staying home.
~q → ~p: If I am not staying home,
then it is not raining.
[p → q] is true.
Then [~q → ~p] is true.
Then [~q → ~p] is true.
Close
Example
If the inverse of the given statement is true,
write a statement that is always true.
If it's raining, then I'm staying home.
Solution write a statement that is always true.
If it's raining, then I'm staying home.
pq
If it's raining, then I'm staying home.
~p → ~q: ( o ) - [1]
q → p: ( o ) - [2]
If I'm staying home, then it's raining.
If it's raining, then I'm staying home.
~p → ~q: ( o ) - [1]
q → p: ( o ) - [2]
If I'm staying home, then it's raining.
[1]
The inverse [~p → ~q] is true.
[2]
Then the converse [q → p] is true.
Close
Law of Detachment
Law
p → q( o )
p( o )
q( o )
If [p → q] and [p] are true,p( o )
q( o )
then [q] is true.
Example
If the given statements are all true,
write a statement that is always true.
If it's raining, then I'm staying home.
It's raining.
Solution write a statement that is always true.
If it's raining, then I'm staying home.
It's raining.
pq
If it's raining, then I'm staying home.
p
It's raining.
p → q( o )
p( o )
q( o )
I'm staying home.
If it's raining, then I'm staying home.
p
It's raining.
p → q( o )
p( o )
q( o )
I'm staying home.
Close
Law of Syllogism
Law
p → q( o )
q → r( o )
p → r( o )
If [p → q] is true and [q → r] is true,q → r( o )
p → r( o )
then [p → r] is true.
Example
If the given statements are all true,
write a statement that is always true.
If it's raining, then I'm staying home.
If I'm staying home, then I'm listening to music.
Solution write a statement that is always true.
If it's raining, then I'm staying home.
If I'm staying home, then I'm listening to music.
pq
If it's raining, then I'm staying home.
qr
If I'm staying home, then I'm listening to music.
p → q( o )
q → r( o )
p → r( o )
If it's raining, then I'm listening to music.
If it's raining, then I'm staying home.
qr
If I'm staying home, then I'm listening to music.
p → q( o )
q → r( o )
p → r( o )
If it's raining, then I'm listening to music.
Close
Biconditional
Definition
p ↔ q
A biconditional is the conjunction of a conditional and its converse.[p → q] ∧ [q → p]
It's written and read as
[p if and only if q], [p iff. q].
Truth Value
p → q | q → p | p ↔ q |
---|---|---|
o | o | o |
o | x | x |
x | o | x |
x | x | x |
if both [p → q] and [q → p] are true.
Example
Solution
Close
p
If 2 is a prime number,
if and only if 2 is an even number.
q
p: ( o )
q: ( o )
p → q: ( o )
q → p: ( o ) - [1]
p ↔ q: ( o )
True
If 2 is a prime number,
if and only if 2 is an even number.
q
p: ( o )
q: ( o )
p → q: ( o )
q → p: ( o ) - [1]
p ↔ q: ( o )
True
[1]
Close
Example
∠A is a right angle iff. m∠A = 90.
Truth value?
Solution Truth value?
pq
∠A is a right angle iff. m∠A = 90.
p → q: ( o ) - [1]
q → p: ( o ) - [2]
p ↔ q: ( o )
True
∠A is a right angle iff. m∠A = 90.
p → q: ( o ) - [1]
q → p: ( o ) - [2]
p ↔ q: ( o )
True
[1]
p → q: If ∠A is a right angle,
then m∠A = 90.
This is true.
then m∠A = 90.
This is true.
[2]
q → p: If m∠A = 90,
then ∠A is a right angle.
This is also true.
then ∠A is a right angle.
This is also true.
a biconditional can be used
to show the definition of something.
Close
Example
x + 2 = 3 iff. x = 1.
Truth value?
Solution Truth value?
pq
x + 2 = 3 iff. x = 1.
p → q: ( o ) - [1]
q → p: ( o ) - [2]
p ↔ q: ( o )
True
x + 2 = 3 iff. x = 1.
p → q: ( o ) - [1]
q → p: ( o ) - [2]
p ↔ q: ( o )
True
[1]
p → q: If [x + 2 = 3], then [x = 1].
This is true.
This is true.
[2]
q → p: If [x = 1], then [x + 2 = 3].
This is also true.
This is also true.
a biconditional can also be used
to show the solution of an equation.
Close
Example
x2 = 4 iff. x = 2.
Truth value?
Solution Truth value?
pq
x2 = 4 iff. x = 2.
p → q: ( x ) - [1]
(∵ x = ±2)
q → p: ( o ) - [2]
p ↔ q: ( x )
False
x2 = 4 iff. x = 2.
p → q: ( x ) - [1]
(∵ x = ±2)
q → p: ( o ) - [2]
p ↔ q: ( x )
False
[1]
p → q: If [x2 = 4], then [x = 2].
If [x2 = 4],
then [x = ±2].
(not x = 2)
So [p → q] is false.
Quadratic Equation
If [x2 = 4],
then [x = ±2].
(not x = 2)
So [p → q] is false.
Quadratic Equation
[2]
q → p: If [x = 2], then [x2 = 4].
This is true.
This is true.
Close