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 sentence
that is either true ( o ) or false ( x ).

Negation

Definition

[~p] is the negation of p.
It means [not p].

Example

Find the negation of the given statement.

5 is a positive number.

Solution

Example

Find the negation of the given statement.

1 + 2 = 0

Solution

Example

Find the negation of the given statement.

2 is not an odd number.

Solution

Truth Value

p	~p
o	x
x	o

p and ~p have the opposite truth values.

Example

p: 5 is a positive number.

Truth value of ~p?

Solution

Example

q: 3 is an even number.

Truth value of ~q?

Solution

Example

r: 2 is a prime number.

Truth value of ~(~r)?

Solution

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

p ∧ q is true
if both p and q are true.

Example

p: 5 is a positive number.
q: 1 + 1 = 3

Truth value of p ∧ q?

Solution

Example

p: 5 is a positive number.
r: 4 > 2

Truth value of p ∧ r?

Solution

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

p ∨ q is true
if either p and q are true.

Example

p: 5 is a positive number.
q: 1 + 1 = 3

Truth value of p ∨ q?

Solution

Example

q: 1 + 1 = 3
s: 2 > 9

Truth value of q ∨ s?

Solution

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

Example

Find the hypothesis and conclusion of the given statement.

If he is not in his room,
then he is playing basketball.

Solution

Example

Find the hypothesis and conclusion of the given statement.

I'm staying home if it's raining.

Solution

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

Example

p: 2 is a prime number.
r: 2 is an odd number.

Truth value of p → r?

Solution

Example

p: 2 is a prime number.
r: 2 is an odd number.

Truth value of r → p?

Solution

Inverse

Definition

~p → ~q

To find the inverse of [p → q],
negate both p and q.

Example

If 2 is a prime number,
then 2 is an odd number.

Inverse?

Solution

Example

If he is not in his room,
then he is playing basketball.

Inverse?

Solution

Converse

Definition

q → p

To find the converse of [p → q],
switch p and q.

Example

If 2 is a prime number,
then 2 is an odd number.

Converse?

Solution

Example

If he is not in his room,
then he is playing basketball.

Converse?

Solution

Contrapositive

Definition

~q → ~p

To find the contrapositive of [p → q],
negate and switch both p and q.

Example

If 2 is a prime number,
then 2 is an odd number.

Contrapositive?

Solution

Example

If he is not in his room,
then he is playing basketball.

Contrapositive?

Solution

Law of Contrapositive

Law

p → q = ~q → ~p

A conditional and its contrapositive
have the same truth value.

Relationship between Conditional, Inverse, Converse, and Contrapositive

The inverse [~p → ~q] and the converse [q → p]
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

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

Law of Detachment

Law

p → q( o )
p( o )
q( o )

If [p → q] and [p] are true,
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

Law of Syllogism

Law

p → q( o )
q → r( o )
p → r( o )

If [p → q] is true and [q → r] is true,
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

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

A biconditional is true
if both [p → q] and [q → p] are true.

Example

2 is a prime number
if and only if 2 is an even number.

Truth value?

Solution

Example

∠A is a right angle iff. m∠A = 90.

Truth value?

Solution

Example

x + 2 = 3 iff. x = 1.

Truth value?

Solution

Example

x² = 4 iff. x = 2.

Truth value?

Solution