Truth Tables
Truth table specifies the truth value of a compound proposition for all possible truth values of its constituent propositions.
Negation(~)
If p is a statement variable, then negation of p,"not p", is denoted as "~p".
It has opposite truth value from p i.e., if p is true, then ~p is false; if p is false, then ~p is true.
Truth Table for ~p
| p | ~p |
| T | F |
| F | T |
Conjunction (∧)
Assume p and q are statements and the conjunction of p and q is "p and q", denoted as "p ∧q".
Note the following regarding the conjunction
- p∧q is true when both p and q are true.
- p∧q is false when either p or q is false, or both are false.
The Truth Table for p∧q is illustrated below:
Truth Table for p∧q
| p | q | p∧q |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
Disjunction (∨) or Inclusive OR
Assume p and q are statements and the disjunction of p and q is "p or q", denoted as "p ∨q".
Note the following regarding the disjunction
- p∨q is true when either p or q is true, or both are true.
- p∨q is false when both p and q are false.
The Truth Table for p∨q is illustrated below:
Truth Table for p∨q
| p | q | p∨q |
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
Exclusive OR
When OR is used in its exclusive sense, the statement "p or q" mean "p or q but not both" or "p or q but not p and q"
which translates into symbols as (p∨q) ∧ ~(p∧q). It is abbreviated as p⊕q or p XOR q.
The Truth Table for p XOR q is illustrated below:
Truth Table for p⊕q
| p | q | p⊕q |
| T | T | F |
| T | F | T |
| F | T | T |
| F | F | F |
Conditional Statements
Consider the following statement: "If you get an A in Computer Science, then I will buy you your favorite phone."
This statement is composed of two simpler statements:
p:"You will get an A in Computer Science"
q:"I will buy you your favorite phone."
The original statement is then saying:
if p is true, then q is true, or simply, if p, then q.
This can also be expressed as: p implies q. It is denoted by p → q.
p → q. is false when p is true and q is false; otherwise it is true.
In p → q., the statement p is called the hypothesis (or antecedent) and q is called the conlcusion (or consequent)
The Truth Table for p → q is illustrated below:
Truth Table for p → q
| p | q | p→q |
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Biconditional
Assume p and q are statement variables, the biconditional of p and q is "p if and only if q".
it is denoted p ↔ q. "if and only if" is abbreviated as iff.
The symbol (double headed arrwo) "↔" is the biconditional operator.
Note the following regarding the biconditional
- p↔q is true when both p and q are true.
- p↔q is false when either p or q is false.
The Truth table for p ↔ q is illustrated below.
Truth Table for p↔q
| p | q | p↔q |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
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Date of last modification: 2024