What is meant by logical equivalence?
Logical equivalence is a type of relationship between two statements or sentences in propositional logic or Boolean algebra. The relation translates verbally into “if and only if” and is symbolized by a double-lined, double arrow pointing to the left and right ( ).
How do you find equivalent logic?
To test for logical equivalence of 2 statements, construct a truth table that includes every variable to be evaluated, and then check to see if the resulting truth values of the 2 statements are equivalent.
What is equivalence in propositional logic?
Two logical expressions are said to be equivalent if they have the same truth value in all cases. Sometimes this fact helps in proving a mathematical result by replacing one expression with another equivalent expression, without changing the truth value of the original compound proposition.
What is logically equivalent to P → Q?
P → Q is logically equivalent to ¬ P ∨ Q . Example: “If a number is a multiple of 4, then it is even” is equivalent to, “a number is not a multiple of 4 or (else) it is even.”
What is equivalence formula?
Two formulas P and Q are said to be logically equivalent if P ↔ Q is a tautology, that is if P and Q always have the same truth value when the predicate variables they contain are replaced by actual predicates. The notation P ≡ Q asserts that P is logically equivalent to Q.
What is a proposition that is always false?
A compound proposition is called a contradiction if it is always false, no matter what the truth values of the propositions (e.g., p A ¬p =T no matter what is the value of p.
What is a proposition that is always true?
Definitions: A compound proposition that is always true for all possible truth values of the propositions is called a tautology. A compound proposition that is always false is called a contradiction. A proposition that is neither a tautology nor contradiction is called a contingency.
What is logical equivalence in discrete mathematics?
Two propositions p and q are logically equivalent if their truth tables are the same. Namely, p and q are logically equivalent if p ↔ q is a tautology. If p and q are logically equivalent, we write p ≡ q.
What is P <-> Q?
P→Q means If P then Q. ~R means Not-R. P ∧ Q means P and Q. P ∨ Q means P or Q. An argument is valid if the following conditional holds: If all the premises are true, the conclusion must be true.
What does P <-> Q mean?
A proposition of the form “if p then q” or “p implies q”, represented “p → q” is called a conditional proposition. For instance: “if John is from Chicago then John is from Illinois”. The proposition p is called hypothesis or antecedent, and the proposition q is the conclusion or consequent.
What is propositional logic example?
Definition: A proposition is a statement that can be either true or false; it must be one or the other, and it cannot be both. EXAMPLES. The following are propositions: – the reactor is on; – the wing-flaps are up; – John Major is prime minister.
How to represent a logic equivalence?
Write the converse and contrapositive of each of the following conditional statements.
How to prove logical equivalence?
– ¬((¬p∧q)∨¬(r∨¬s)). ¬ ( ( ¬ p ∧ q) ∨ ¬ ( r ∨ ¬ s)). – ¬((¬p→ ¬q)∧(¬q → k)) ¬ ( ( ¬ p → ¬ q) ∧ ( ¬ q → k)) (careful with the implications). – (p∧q)→ (p∨q) ( p ∧ q) → ( p ∨ q)
How to solve logical equivalence?
– p∧q ≡ ¬(p → ¬q) p ∧ q ≡ ¬ ( p → ¬ q) – (p → r)∨(q → r)≡ (p∧q)→ r ( p → r) ∨ ( q → r) ≡ ( p ∧ q) → r – q → p≡ ¬p→ ¬q q → p ≡ ¬ p → ¬ q – (¬p → (q∧¬q))≡ p ( ¬ p → ( q ∧ ¬ q)) ≡ p
What does logical equivalence mean?
Logical equivalence occurs when two statements have the same truth value. This means that one statement can be true in its own context, and the second statement can also be true in its own context, they just both have to have the same meaning.