A system that deals with Proposition (statements).
A declarative sentence that is either true or false (but not both) is a proposition.
Also known as a statement.
These sentences would be considered propositions:
- It is Thursday today (true).
- I am 14 years old (false).
- 1 + 1 = 3 (false).
Usually denoted using lowercase letters , , , or .
p = It rained yesterday. q = I am happy.
The truthfulness or falsity of a proposition is called its Truth Value. Denoted by or , or 1 and 0 in computer science.
We can use connectives to change or combine the meaning of propositions. For example, negates the value of p. If it's true, it becomes false and vice versa.
A truth table allows us to consider all possible combinations of proposition logic systems.
For example, consider :
We can use truth tables to help us understand the truth values of other connectives within propositional logic.
An operator that negates a proposition.
- = I will pass my exam.
- = I will NOT pass my exam.
In Boolean Algebra, it's equivalent to
True when p OR q is true.
True only when p AND q is true.
Equivalent to multiplication
If p is true, then q is true.
Think of it as a promise. Only false when promise is broken.
Equivalent to equality check:
p or q but not both.
Also called XOR.
Truth table is opposite of bi-conditional.
A statement that is always true.
For example: is always true.
A formula that is true in at least one scenario.
A formula that is never true.
Also called "inconsistent".
If two formula are equivalent if they have identical truth tables.
means A and B have the same Truth Table.
Note: equivalence is relation, not connective.
Can prove De Morgan's Laws using a truth table.