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# 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
[~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
ox
xo
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

pqp ∧ q
ooo
oxx
xox
xxx
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

pqp ∨ q
ooo
oxo
xoo
xxx
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

pq
[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,
Solution

### Example

Find the hypothesis and conclusion of the given statement.

I'm staying home if it's raining.
Solution

pqp → q
ooo
oxx
xoo
xxo

### 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 [pq],
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,

Inverse?
Solution

## Converse

### Definition

qp
To find the converse of [pq],
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,

Converse?
Solution

## Contrapositive

### Definition

~q~p
To find the contrapositive of [pq],
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,

Contrapositive?
Solution

## Law of Contrapositive

### Law

pq = ~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

pq( 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

pq( o )
qr( o )
pr( 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

pq
A biconditional is the conjunction of a conditional and its converse.
[pq] ∧ [qp]

[p if and only if q], [p iff. q].

### Truth Value

p → qq → pp ↔ q
ooo
oxx
xox
xxx
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