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# logics

Logic notations
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## Logic

 $\inline&space;&space;\displaystyle&space;X,\,&space;Y,\,&space;Z,\,&space;\ldots$ Various propositions that are assumed to be either true or false (not both), e.g. $\inline&space;X$: "the dog is an animal", $\inline&space;Y$: "$\inline&space;5$ is greater than $\inline&space;8$". A true proposition is marked with a $\inline&space;T$ while a false one is marked with $\inline&space;F$. $\inline&space;&space;X&space;\wedge&space;Y$ This is a proposition called the logical conjuction of $\inline&space;X$ and $\inline&space;Y$. It is true when both propositions are true, and false otherwise. $\begin{array}{c|c|c}&space;&space;X&space;&&space;Y&space;&&space;X&space;\wedge&space;Y&space;\\&space;&space;\hline&space;&space;T&space;&&space;T&space;&&space;T&space;\\&space;&space;F&space;&&space;T&space;&&space;F&space;\\&space;&space;T&space;&&space;F&space;&&space;F&space;\\&space;&space;F&space;&&space;F&space;&&space;F&space;\\&space;&space;\end{array}$ $\inline&space;&space;\displaystyle&space;X&space;\vee&space;Y$ This is a proposition called the logical disjunction of $\inline&space;X$ and $\inline&space;Y$. It is true when either one of the propositions are true, and false otherwise. $\begin{array}{c|c|c}&space;&space;X&space;&&space;Y&space;&&space;X&space;\vee&space;Y&space;\\&space;&space;\hline&space;&space;T&space;&&space;T&space;&&space;T&space;\\&space;&space;F&space;&&space;T&space;&&space;T&space;\\&space;&space;T&space;&&space;F&space;&&space;T&space;\\&space;&space;F&space;&&space;F&space;&&space;F&space;\\&space;&space;\end{array}$ $\inline&space;&space;\displaystyle&space;\overline{X}$ This is a proposition called the logical negation of $\inline&space;X$. It is true when $\inline&space;X$ is a false proposition, and false if $\inline&space;X$ is true. $\begin{array}{c|c}&space;&space;X&space;&&space;\overline{X}&space;\\&space;&space;\hline&space;&space;T&space;&&space;F&space;\\&space;&space;F&space;&&space;T&space;\\&space;&space;\end{array}$ $\inline&space;&space;\displaystyle&space;P(x),\,&space;Q(x,y),\,&space;\ldots$ These are called predicates and they are used to express propositions about various objects, hence they can either be true or false depending on the object in question. For example if $\inline&space;P(x)$ is the predicate "$\inline&space;x$ is a prime number", then by replacing $\inline&space;x$ with the value of $\inline&space;13$ we obtain $\inline&space;P(13)$: "$\inline&space;13$ is a prime number", thus a true proposition. However if we let $\inline&space;x$ be $\inline&space;224$, then the proposition $\inline&space;P(224)$: "$\inline&space;224$ is a prime number" is obviously false. $\inline&space;&space;\displaystyle&space;\exists&space;x&space;\in&space;A:&space;P(x)$ This proposition expresses the fact that there exists an object $\inline&space;x$ from a set $\inline&space;A$ of various objects, such that the predicate $\inline&space;P(x)$ becomes true. For example if the set of objects is $\inline&space;A&space;=&space;\{\,1,\,&space;2,\,&space;3,\,&space;4\,\}$ and the predicate is $\inline&space;P(x)$: "$\inline&space;x$ is even" then the following proposition is true, $\inline&space;\exists&space;x&space;\in&space;A:&space;P(x)$. Obviously for $\inline&space;x$ equal to $\inline&space;4$ the predicate $\inline&space;P(x)$ becomes true. The symbol $\inline&space;\exists$ is called the existential quantifier. $\inline&space;&space;\displaystyle&space;\forall&space;x&space;\in&space;A:&space;P(x)$ This proposition expresses the fact that for all objects $\inline&space;x$ from a set $\inline&space;A$ of various objects, the predicate $\inline&space;P(x)$ is true. For example if the set of objects is $\inline&space;A&space;=&space;\{\,$ dog, cat, rabbit, fox $\inline&space;\,\}$ and the predicate is $\inline&space;P(x)$: "$\inline&space;x$ is an animal" then the following proposition is true, $\inline&space;\forall&space;x&space;\in&space;A:&space;P(x)$. Obviously for any object $\inline&space;x$ in the set $\inline&space;A$ the predicate $\inline&space;P(x)$ is true. The symbol $\inline&space;\forall$ is called the universal quantifier.