Conditional Statement

Conditional Statement is a #202204281244 which states that if one thing is true, then some other thing should be also true. It usually uses the phrase “if … then”.

For example: If primate is mammal, then human is mammal. This can be denoted using logical operator#, \(p \rightarrow q\). \(p\) is called a hypothesis or antecedent whereas \(q\) is called the conclusion or consequent. The hypothesis of the statement could be false, however it doesn’t imply that the statement is false. On the contrary, the conclusion could still be true, which we call it as vacuously true or true by default. This could be seen from its 202205061151#. If the hypothesis is true, then the conclusion must be true, otherwise it is a false statement.

If both hypothesis and conclusion are in their negation form, it is the inverse of the original conditional statement, shown as \(\sim p \rightarrow \sim q\). When the conclusion and the hypothesis swap their places in a statement, as in \(q \rightarrow p\), it is a converse of the conditional statement. The contrapositive of a conditional statement can be formed when the hypothesis and conclusion in converse of the conditional statement are negated, denoted \(\sim q \rightarrow \sim p\).

The contrapositive of a conditional statement is logically equivalence# to the conditional statement itself. Both converse and inverse form of the conditional statement is not logically equivalence to it (the conditional statement), but they are logically equivalence to themselves (converse and inverse).

There is another form of Conditional Statement which is more restrictive, the Biconditional. It is denoted \(p \leftrightarrow q\) which means \(p\) if and only if \(q\). For example: Human is mammal if, and only if primate is mammal. We can either say that “If human is mammal, then primate is mammal” or “If primate is not mammal, then human is not mammal”. Sometimes, the phrase “if and only if” will be abbreviated to “iff”. Note that the statement \(P\) and \(Q\) are logically equivalence# if and only if \(P \leftrightarrow Q\) is a tautology.

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Universal Statement

Just like 202205062055, Universal Statement could be said is vacuously true or true by default if \(Q(x)\) is false for every element in the set. In other word, if the negation of the Universal Statement is false, then the Universal Statement is true, even if there is no element in the set. Consider the statement “All books on the shelf has a cover”. If none of them are on the shelf, then the statement is true, as the negation of it “There is at least one book on the shelf which doesn’t have a cover” is false, as there is no books on the shelf at all.
Universal Conditional Statement

Some 202204281244 can express both property of #202204281245 and #202205062055 which make it a #202205061208, named Universal Conditional Statement.
Transitivity

It is commonly used in #math especially if there are chain of #202205062055. For example, if the number is prime, then the number can only be divided by itself, and if the number can only be divided by itself, then it share no greatest common factor (GCF) to other prime number.
The Laws of Logic

Following shows the laws applied on #202205062055:
Necessary and Sufficient Condition

If a condition is both necessary and sufficient, this implies biconditional statement#. The statement “\(r\) if, and only if \(s\)” expresses such condition.

Necessary Condition is a condition where one 202204281244 solely depends on another statement. This could be expressed in 202205062055# as “if not \(r\), then not \(s\)”. “If \(s\) then \(r\)” means the same thing.
Modus Tollens

Note the similarity with the contrapositive of the 202205062055. Therefore, we can know that Modus Tollens is logically equivalent to 202206101747.
Mathematical Statement

202205062055#
Denying The Antecedent

Denying The Antecedent or Inverse Error is a #202206172053 where one draw their conclusion from premises by replacing the premise \(p \rightarrow q\) by its inverse. From 202205062055# we know that the inverse of the conditional statement is not logically equivalence# to itself. The following shows its form:
Cyclic Code

Due to the nature of Cyclic Code, we can state that a binary polynomial of order \(n-1\) or less is a code word polynomial if and only if it is a multiple of \(g(D)\)#. Multiply a polynomial with \(D^i\) modulus with \(D^n + 1\) can be viewed as a \(i\) right cyclic shift.
Affirming The Consequent

Affirming The Consequent or Converse Error is a kind of #202206172053 where one draw a conclusion from the premises as if it is conversed. As we can see in 202205062055#, the converse of the conditional statement is not logically equivalence# to itself. The form could be seen below:

#logic