# Truth tables tautologies and logical equivalence pdf

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Published: 04.05.2021  A tautology is a proposition that is always true, regardless of the truth values of the propositional variables it contains. A proposition that is always false is called a contradiction. A proposition that is neither a tautology nor a contradiction is called a contingency.

## Discrete Mathematics Study Center

A tautology is a proposition that is always true, regardless of the truth values of the propositional variables it contains. A proposition that is always false is called a contradiction.

A proposition that is neither a tautology nor a contradiction is called a contingency. This claim is always true. This is clearly impossible. We can use a truth table to verify the claim. It is easy to verify with a truth table. We can also argue that this compound statement is always true by showing that it can never be false.

Thus, [eqn:tautology] cannot be false, it must be a tautology. We list the truth values according to the following convention. So we split the upper half of the second column into two halves, fill the top half with T and the lower half with F. Likewise, split the lower half of the second column into two halves, fill the top half with T and the lower half with F.

What we are saying is, they always produce the same truth value, regardless of the truth values of the underlying propositional variables. The truth tables for a and b are depicted below. Properties of Logical Equivalence. Be sure you memorize the last two equivalences, because we will use them frequently in the rest of the course. These properties are not easy to recall. Instead of focusing on the symbolic formulas, try to understand their meanings.

Let us explain them in words, and compare them to similar operations on the real numbers,. Give a logical explanation as well as a graphical explanation.

Its negation yields two open intervals. Their graphical representations on the real number line are depicted below. Take note of the two endpoints 2 and 3. They change from inclusion to exclusion when we take negation.

We need to apply the distributive law twice. The complete solution is shown below. This kind of proof is usually more difficult to follow, so it is a good idea to supply the explanation in each step. Answer We can use a truth table to verify the claim. In other words, show that the logic used in the argument is correct. Summary and Review Two logical statements are logically equivalent if they always produce the same truth value.

Exercises 2. Which ones are tautologies? Do not delete this text first. ## Discrete Mathematics Study Center

A less abstract example is "either the ball is green, or the ball is not green". This would be true regardless of the color of the ball. The philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in , borrowing from rhetoric , where a tautology is a repetitive statement. In logic, a formula is satisfiable if it is true under at least one interpretation, and thus a tautology is a formula whose negation is unsatisfiable. Unsatisfiable statements, both through negation and affirmation, are known formally as contradictions. A formula that is neither a tautology nor a contradiction is said to be logically contingent. Such a formula can be made either true or false based on the values assigned to its propositional variables.

Could both trolls be knights? Recall that all trolls are either always-truth-telling knights or always-lying knaves. A proposition is simply a statement. Propositional logic studies the ways statements can interact with each other. It is important to remember that propositional logic does not really care about the content of the statements.

Truth Tables How can we determine the truth value of compound propositions? Adding more variables means adding more rows. Then fill the second column by repeating this pattern in each half, and so on. This is an easy way to guarantee all possibilities are covered. If a statement is neither a tautology nor a contradiction, then the truth values do alter the outcome and we say that the statement is a contingency. Therefore, the proposition is a tautology. ## Discrete Mathematics Study Center

Predicate Calculus An assertion in predicate calculus isvalidiff it is true Discrete Mathematics. An Example from Calculus Express that the limit of a real-valued function f at point a is L.

Logic is a proper or reasonable way of thinking about or understanding something, and the science that studies the formal processes used in thinking and reasoning. You can't get very far in logic without talking about propositional logic also known as propositional calculus. A proposition is a declaritive sentence a sentence that declares a fact that is either true or false. Examples of propositions: Tallahassee is the capital of Florida Washington D. We use letters to denote propositional variables , similar to how letters can represent numbers.

Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. Connect and share knowledge within a single location that is structured and easy to search. In general, in propositional classical logic which is the logic where truth tables make sense , a standard way to prove that a formula is a tautology without using truth table is:. Note: You are looking at a static copy of the former PineWiki site, used for class notes by James Aspnes from to

### propositional calculus in discrete mathematics pdf

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