Logic

converse

he converse is not logically equivalent to the original statement. The converse of if p then q is q then p.

Proposition Type

- simple - combined A sentence containing a single proposition is not hard to evaluate to true or false. But when it's combined with logical operators, it can be quite a different story. Logical operators are used to make compound statements or compound propositions. To help with evaluating these more complex statements, we can dissect the sentence into predicates and assign each predicate a different variable.

Operator

A compound proposition is created by connecting individual propositions using logical operators. AND is denoted with an upside down v symbol, such as p AND q. The truth value of a compound proposition using the AND operator is only true when both propositions are true. OR is denoted using a v symbol, such as p OR q. The truth value of a compound propositionusing the OR operator is always true except when p and q are both false. If a statement has logical operators, the order of precedence is the NOT operator first, then AND, and finally OR.

Predicate

A logical statement whose truth value is a function of one or more variables is called a predicate. A logical statement whose truth value is a function of one or more variables is called a predicate. It might be a function which takes more than one variable.

Quantifier

A predicate can also be defined as a proposition by adding a quantifier. There are two types of quantifiers. The universal quantifier, which is read as for all. And an existential quantifier, which represents there exists. Notice the symbols. A universal quantifier looks like an upside down capital A. The existential quantifier looks like a backwards capital E.When a predicate has a quantifier, if the domain is small, it is often easy to prove whether or not this is true using the method of exhaustion. When a predicate has a quantifier, if the domain is small, it is often easy to prove whether or not this is true using the method of exhaustion. When a predicate has a quantifier, if the domain is small, it is often easy to prove whether or not this is true using the method of exhaustion. The difference in existential is I just have to find one value that matches the criteria

Theorem

A theorem is a statement that can be proven true or false. A proof consists of a series of steps, each of which follows logically from assumptions or from previously proven statements whose final step should result in a statement of the theorem being proven.

Logical Equivalence

Although two statements might have very different semantic meaning, such as, dogs bark and cats meow, this can actually be logically equivalent to, the sky is blue and the grass is green. At first, this might sound absurd. But the reality is that both statements, on either side of the and operator, are true. So the overall statement is true. And if the overall statements match, in this case, both true, then they are considered logically equivalent. Truth tables are used to determine logical equivalence. By using the canonical form, we can easily determine if the conclusions match, without reordering the rows. To prove that two statement forms are not logically equivalent, we only need to find one row that is different.

Argument (inference)

An argument consists of a sequence of propositions, called hypotheses. They're followed by a final proposition, called the conclusion. All propositions are either true or false. But when dealing with arguments, it's important to note that an argument is either valid or invalid. Let's take a look at determining argument validity. An argument is valid if the conclusion is true whenever all the hypotheses are true.Otherwise, it's considered invalid. An argument form contains one or more hypotheses and one conclusion,

Proofs by Cases

Another way to solve proofs is by cases. Say I wanted to prove the statement that for all integers n n squared plus n plus one is odd. The proof must be completed by starting with n as an even integer. This is case one. Then we start with n as an odd integer. This is case two. If both cases are true, we have proven our original theorem.

Axioms

Axioms are statements assumed to be true, or previously proven theorems.

Biconditional proofs

Biconditional statements are written as p with a double arrow q. When you see that, it means p, if and only if, q. The biconditional statement is true when both p and q have the same truth value and false if they are different. The only time that a biconditional statement is false is when they don't match.

Conditional Statement Rule

Conditional statement is logically equivalent to to p AND NOT q. p --> q = !p v q = p ^!q , p --> q = !q --> !p

Conditional statements

Conditional statements, or conditional propositions, form the basis for all decision making in computer software. Back to conditional propositions. A conditional statement is read as 'if this, then that'. Conditional statements are written using an arrow, such as if p then q. p represents the premises (Hypothesis) and q is the conclusion. conditional statement is false only when the hypothesis is true and the conclusion is false. All other values result in the statement being true. At first, this might be strange and hard to comprehend. it doesn't matter whether it's true or false because the entire statement is going to be true. There are three variations of the conditional statements, contrapositive, inverse, and converse.

Direct Proof

In a direct proof, you start by identifying the hypotheses. For example, you start with suppose.Then, identify the conclusion that you will prove. Finally, reason through the steps to prove your conclusion. Don't forget to identify your reason for each step using assumptions or previously proven statements.

Proposition

In the study of valid reasoning, statements, also called propositions, are sentences that are either true, or false. It is important to understand that not all sentences are propositions. In other words, not all sentences, can be evaluated to true or false. Exclamatory and questioning sentences are not usually propositions. This provides a perfect correlation with computing, as true or false answers are represented as a one or a zero, on or off. So remember, to study valid reasoning, the communication must be clear, concise, and unambiguous.

De Morgan's Law

It specifically addresses how to negate a conjunction or disjunction statement. In other words, how to negate an and or an or.

Mathematical Induction

One of the most important types of proof in discrete mathematics is called mathematical induction. This process allows you to verify a given theorem. Induction is the proof technique that is especially useful for proving statements about elements in a sequence. The two components of the inductive proof are first, identifying the base case, which establishes that the theorem is true for the first value in the sequence. Next, you identify the inductive step, which establishes that if the theorem is true for k, then the theorem holds true for k plus one.

Contradiction

Proof by contradiction is another type of indirect proof, and requires a counter-example.

Proof categories

Proofs fall into two categories, direct and indirect proofs.

Tautology

Tautology a compound proposition is considered a tautology if the proposition is always true, regardless of the truth value of the individual propositions that occur in it.A simple example is, p or not p.This is always true.

contrapositive

The contrapositive is the only one that is considered logically equivalent to the original statement. We start with if p then q, and this is logically equivalent toNOT q, then NOT p. So you negate both sides and switch the values.

Domain of function

The domain of the function is the set of all possible values for the variables. This is what determines if it is true or false, and it can be finite or infinite.

inverse

The inverse is not logically equivalent to the original statement. If p then q becomes NOT p, then NOT q. And the last one is called the converse.

Contradiction

The opposite is a contradiction. A compound proposition is a contradiction if the proposition is always false, regardless of the truth value of the individual propositions that occur in it. A simple example is p and not p. This is always false.

Truth Table

Truth tables can be very useful in evaluating complex logical statements. Here is a guide to creating these types of truth tables. First, you calculate the number of required rows. As you can see here, I have three variables, so I needed eight rows. The number of rows is represented by two raised to the power of the number of variables, so two cubed is eight. Next, I set n equal to the number of rows divided by two. So I'm going to say n is equal to four. I fill in the first column with true until I get to n rows.

rules of inference

When evaluating arguments for validity, there are several rules of inference that can make the process much easier. The idea is to take the original argument and label each predicate with a unique variable. Then rewrite the problem using the variables, the logical connectors, and conditional statements such as and, or, if then, not, et cetera. Now we can look at the rules of inference to find a match to our format. If we find a match, we know this is a valid argument.

Indirect Proof

With an indirect proof, instead of proving that the conclusion is true, you start by assuming that the opposite is true. Then, using deductive reasoning to lead to a contradiction, you can prove the original hypotheses is true.

Logic

is the study of the principles of valid reasoning and inference