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The proposition ‘No historians are non-mathematician’ is equivalent to which of the following proposition?

1. All historians are mathematicians
2. No historians are mathematicians
3. Some historians are mathematicians
4. Some historians are not mathematicians

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+1 vote

Let $x$ : A person

$h(x)$: $x$ is a historian  and

$m(x)$: $x$ is a mathematician

No historians are non-mathematicians $\equiv$ There does not exist $x$ such that $x$ is historian and not a mathematician.$\equiv \sim \exists x(h(x)\wedge \sim m(x)) \equiv \forall x(\sim h(x) \vee m(x))$

1. All historians are mathematicians $\equiv$ For all $x$, if $x$ is a historian, then it has to be  a mathematician $\equiv \forall x(h(x)\rightarrow m(x))\equiv \forall x(\sim h(x) \vee m(x))$
2. No historians are mathematicians $\equiv$ There does not exist $x$ such that $x$ is a historian and $x$ is a mathematician$\equiv \sim \exists x( h(x) \wedge m(x)) \equiv \forall x(\sim h(x) \vee \sim m(x))$
3. Some historians are mathematicians $\equiv$ There exist some $x$ such that $x$ is a historian and also a mathematician$\equiv \exists x( h(x) \wedge m(x))$
4. Some historians are not mathematicians $\equiv$ There exist some $x$ such that $x$ is a historian and also not a mathematician$\equiv \exists x( h(x) \wedge \sim m(x))$

$\therefore$ Option $1.$ is correct.

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