Math Thinking
Sets
N : the set of all natural numbers (i.e., the numbers 1, 2, 3, etc.)
Z : the set of all integers (0 and all positive and negative whole numbers)
Q : the set of all rational numbers (fractions)
R : the set of all real numbers
Implication
Implication has a truth and a causation part.
The truth part is called the conditional part.
Implication = Conditional + causation
In math we focus only on the conditional part.
A
B
A= Antecedent , B= consequent
Defines the truth of A implies B in terms of truth or falsity of A,B.
P: n >7
Q: N^2^>40
(second row of truth table ) T F T = following the real meaning of implication if P would imply Q then if P would be true Q would have to be true also therefore if P is true and Q is false the implication is False.
(third and fourth row): if the antecedent is False then it is easier to look at a negated implication instead:
P does not imply Q: Is true if even though P is true Q is nevertheless false. In all other cases it will be false.
Therefore for non negated implication for all other cases the implication will be True
P implies Q is equivalent to (⌐ P) Or Q
| P | ⌐P | Q | P implies Q | ⌐P Or Q |
|---|---|---|---|---|
| T | F | T | T | T |
| T | F | F | F | F |
| F | T | T | T | T |
| F | T | F | T | T |
P does not imply Q is equivalent to P And ⌐Q
| P | Q | ⌐Q | P implies Q | P does not imply Q | P And ⌐Q |
|---|---|---|---|---|---|
| T | T | F | T | F | F |
| T | F | T | F | T | T |
| F | T | F | T | F | F |
| F | T | F | T | F | F |
Equivalence
P and Q is said to equivalent (logically) if each implies the other.
Biconditional P ⇔ Q is an abbreviation of (P ⇒Q) and (Q ⇒ P)
| P | Q | P ⇔ Q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
One way to show that two statements are equivalent is to show that they have the same truth table.
A:(P And Q) OR (¬ P) is equivalent to B: P ⇒ Q
| P | Q | P And Q | ¬ P | (P And Q) OR (¬ Q) | P ⇒ Q (known) |
|---|---|---|---|---|---|
| T | T | T | F | T | T |
| T | F | F | F | F | F |
| F | T | F | T | T | T |
| F | F | F | T | T | T |
Equivalence and Implication in Language
The following all means "P implies Q":
-
If P then Q
-
P is sufficient for Q
-
P only if Q
-
Q if P
-
Q whenever P
-
Q is necessary for P
The following all means "P is equivalent to Q":
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P is necessary and sufficient for Q (statements 2. and 6. from above combined)
-
P if and only if Q (statements 3. and 4. From above combined) (abbrev. Iff)
Quantifiers
There exists... (Existential quantifier ∃)
There is an object X having property P.
There exists a real number x such that x^2^ + 2x + 1 = 0
∃x[x^2^ + 2x + 1 = 0] find example that solves equation (-1)
∃x[x^3^ + 3x + 1 = 0]
For all...(universal quantifier)
∀ x "For all x it is the case that..."
Combinations of quantifiers
There is no largest natural number
(∀ m ∈ ℕ) (∃ n ∈ ℕ) (n > m)
For all m in the set of natural numbers there exists a natural number n such that n is greater than m
Swapping quantifiers changes meaning
(∃ n ∈ ℕ) (∀ m ∈ ℕ) (n > m)
There is a natural number n that is the biggest natural number.
Let A(x) be some property of x (e.g. x is a real root of the equation x^2^*2x+1=0)
We show that ⌐(∀x A(x)) is the same as ∃x(⌐A(x))
It's not the case that for all x A(x) is true is the same as saying that there is an x where A(x) is not true