Logical Operators

In Discrete Mathematics, the use of propositions alone is too limiting to describe complex and dynamic logical scenarios. Operators make it possible to combine multiple propositions:

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Negation operator

Taking for example the propositions $r,p,q$ , we can negate all of them using the negation operator, noted as $\neg$, $\sim$, or $\stackrel{ˉ}{}$ . It’s possible to use it with any proposition to declare the opposite of it. If the $r$ is true, then $\neg r$ is false. One could consider $\neg r$ as ‘not $r$‘.

For already false propositions, when negating it, they are declared as true propositions, e.g. :

• $p=\text{’Today is NOT going to rain.’}$
• $\neg p=\text{’Today is going to rain.’}$

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Conjunction operator

Conjunctions evaluates the truth value of both propositions, similar to the AND logical operator. In this case, the used notation is $\wedge$. Let $p$ and $q$ as propositions, $p\wedge q$ will be the conjunction of both propositions, interpreted as $p$ and $q$.

A conjunction will be true only if both propositions are true, otherwise the proposition can be considered as false.

• $p=\text{true}$ and $q=\text{false}$. $p\wedge q=\text{false}$
• $p=\text{false}$ and $q=\text{false}$. $p\wedge q=\text{false}$
• $p=\text{true}$ and $q=\text{true}$. $p\wedge q=\text{true}$

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Disjunction operator

Disjunction also evaluates two propositions, but it’s similar to the OR logical operator. It declared as true if at least one of the propositions are true. In this case, the used notation is $\vee$. e.g. :

• $p=\text{true}$ and $q=\text{false}$. $p\vee q=\text{true}$
• $p=\text{false}$ and $q=\text{false}$. $p\vee q=\text{false}$
• $p=\text{true}$ and $q=\text{true}$. $p\vee q=\text{true}$

Exclusive Disjunction

There’s also the exclusive disjunction, which will evaluate the statement as false if both propositions are true, similar to the XOR logical operator. In this case, the used notation is $\oplus$. e.g. :

• $p=\text{true}$ and $q=\text{false}$. $p\oplus q=\text{true}$
• $p=\text{false}$ and $q=\text{false}$. $p\oplus q=\text{false}$
• $p=\text{true}$ and $q=\text{true}$. $p\oplus q=\text{false}$

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Conditional Operator

Conditional operators can also be used to connect propositions and create complex logic and relationships between different propositions. Also known as implications.

Given two propositions $p$ and $q$, the conditional proposition $p\to q$ declares “if $p$, then $q$“. In other words: $p$ implies $q$, where $p$ is a premise and $q$ is the conclusion. $p\to q$ is only false when $p$ is true and $q$ is false, and is true otherwise.

• $p=\text{true}$ and $q=\text{false}$. $p\to q=\text{false}$
• $p=\text{false}$ and $q=\text{false}$. $p\to q=\text{true}$
• $p=\text{true}$ and $q=\text{true}$. $p\to q=\text{true}$
• $p=\text{false}$ and $q=\text{true}$. $p\to q=\text{true}$

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Biconditional Operator

The biconditional statement ”$p$ if and only if $q$”, denoted as $p\iff q$, is only true when $p$ and $q$ carry the same truth value, and is false otherwise. Also abbreviated as ”$p$ iff $q$“.

In other words, the biconditional operator evaluates if both $p$ and $q$ share the same value.

• $p=\text{true}$ and $q=\text{false}$. $p\iff q=\text{false}$
• $p=\text{false}$ and $q=\text{false}$. $p\iff q=\text{true}$
• $p=\text{true}$ and $q=\text{true}$. $p\iff q=\text{true}$
• $p=\text{false}$ and $q=\text{true}$. $p\iff q=\text{false}$