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Logic and Statements

Statement - A mathematical phrase that is true or false

Examples:

  • P: Today is Monday. –> false
  • Q: The integer 5 is even. –> false
  • R: The integer 0 is even. –> true

Unrelated: This shows how it is possible to proove that 0 is even. n is even if and only if n = 2k for some integer k. 0 = 2 * 0 ***

The following examples of P, Q and R display their meaning in plain terms.

  • P v Q is true –> “At least one of these is true.”
  • Q ^ R is true –> “Both of these are true.”
  • P ⊕ R is true –> “One of these is true but not both.”

Laws

Demorgan’s law:
~(PvQ) ≡ ~P ^ ~Q
~(P^Q) ≡ ~P v ~Q

Examples:

~(1 ≤ n < 3)
<=========○—–●=========>
1 3
Also: n < 1 OR n ≥ 3

1 ≤ n < 3
<———●=====○———>
1 3

1.) Negate the following statements.
The earth has two moons or 3 + 5 < 2
The earth does not have two moons and 3 + 5 ≥ 2

2.) Negate the following statements.
A stack overflow has occured and the sector size is not supported.
A stack overflow has not occured and the sector size is supported.

3.) Negate the following statements.
x is positive or y is negative.
x is not positive or y is not negative.

Conditional Statements (Implications)

P –> The hypothesis
Q –> The conclusion

If P then Q

If n is even, then n2 is even
Hypothesis: n is even
Conclusion: n2 is even

There is many ways in which implications can be worded. Some examples are:

P => Q
If P, then Q
P implies Q
P only if Q
Q if P
P is sufficient for Q
Q is necessary for P

Negating an if then statement (negating an implication)

~(P => Q) ≡ ~(~P v Q)
~(~P v Q) ≡ p ^ ~Q

Examples:

Negate the following.
If x > y, then x2 > y2
x > y and x2 ≤ y2
Note: Do not write the If in the negation.

Negate the following.
If rs = 0, then r = 0 or s = 0
rs = 0 and r != 0 and s != 0
Notice: The negation results in the form P ^ ~Q.

Contrapositive and converse statements

If P, then Q.
Contrapositive: If ~Q, then ~P.
Converse: If Q, then P.

If T is equalateral, then T is isosceles.
Note: isosceles defines a triangle in which two sides are congruent.
Contrapositive: If T is not isoscelesm then T is not equalateral.
Converse: If T is isosceles, then T is equalateral.
Note: The contrapositive of this statement is true while the converse of this statement is false.
The contrapositive will always be logically equivilant to the origional statement.

Biconditional statements

P <=> Q
“P if and only if Q”
“If p, then Q and if Q then P”

Examples:

If ab is even, then a is even and b is even.
False, 2 * 3 = 6, 6 is even but 2 is even and 3 is not even.

If ab is even then, a is even or b is even.
True
Converse: If a is even or b is even, then ab is even.
True
Biconditional: ab is even if and only if a is even or b is even.
Note: The statement and the converse are not always both true, but when they are it means the biconditional is true.

Set notation

  • ℕ is the set of natural numbers.
  • Mathemeticians and computer scientists define the set of natural numbers diferently.
  • Mathemeticians: ℕ = {1, 2, 3, …}

  • Computer scientists: ℕ = {0, 1, 2, …}

  • ℤ is the set of integers.

  • ℤ = {…, -3, -2, -1, 0, 1, 2, 3, …}

  • ℝ is the set of real numbers.

  • ℝ = {1, 12.38, -0.8625, 3/4, √2, 198}

  • ℚ is the set of rational numbers.
  • ℚ = { a/b a, b ∈ ℤ, b != 0}
  • 3/7 ∈ ℚ
  • -4000/201 ∈ ℚ
  • 4 = 4/1 ∈ ℚ
  • π ∉ ℚ

Note: The symbol ‘∈’ is pronounced is a subset of.