Statement - A mathematical phrase that is true or false
Examples:
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.
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.
P –> The hypothesis
Q –> The conclusion
If P then Q
If n is even, then n^{2} is even
Hypothesis: n is even
Conclusion: n^{2} 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
~(P => Q) ≡ ~(~P v Q)
~(~P v Q) ≡ p ^ ~Q
Examples:
Negate the following.
If x > y, then x^{2} > y^{2}
x > y and x^{2} ≤ y^{2}
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.
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.
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.
Computer scientists: ℕ = {0, 1, 2, …}
ℤ = {…, -3, -2, -1, 0, 1, 2, 3, …}
ℝ = {1, 12.38, -0.8625, 3/4, √2, 198}
ℚ = { a/b | a, b ∈ ℤ, b != 0} |
Note: The symbol ‘∈’ is pronounced is a subset of.