I have been trying to apply De morgan's law in Logic gates, and realized I am not quite sure if I can use it on my own if given a random problem, which clearly means I dont understand it or connect to it in real life. So could anyone explain, where it is used in a real-life scenario, and also why it is used, and where and when we can use them.
for example : (A and B)' = A' OR B' , I mean how do I connect this in real life?
On
$(A$ and $B$)' means that it is not the case that we have both $A$ and $B$. So it must be the case that we don't have one of those which means either we don't have $A$ or we don't have $B$.
If it is not the case that $A$ and $B$ are both boys, then at least one of them must be a girl.
On
De Morgan's laws with sets.
$$x \notin A\cap B$$ means that $$x \notin A \text{ or } x\notin B$$
and
$$x\notin A\cup B$$ means $$x \notin A \text{ and } x\notin B.$$
If a country is not a permanent member of the security council, then it is not France, and it is not USA and it is not China and it is not Russia and it is not UK.
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"I didn't go to Boston or I didn't go to Chicago" means I didn't go to both Boston and Chicago (though I might have gone to one of them). That is $\neg(B)\vee \neg(C)\equiv \neg(B\wedge C)$
"I don't like spiders and I don't like cockroaches" means I like neither spiders nor cockroaches. That is $\neg S \wedge \neg C \equiv \neg(S\vee C)$
On
They're baked into the English language. Consider the statements
I don't own a car and I don't own a bicycle.
I don't own a car or a bicycle.
I don't own a car and a bicycle.
The first two express the same idea, and are of the forms $\neg A \land \neg B$ and $\neg(A\lor B)$ respectively. This equivalence is one of De Morgan's laws.
The third statement is of the form $\neg (A\land B)$ and isn't equivalent to the first two, as it's possible I don't own a bicycle but do own a car (or vice versa) in addition to the possibility of owning neither. What I just wrote out is $\neg A\lor\neg B$, and De Morgan's law correctly states that it is equivalent to $\neg (A\land B)$
Let's say there's some number $x$ you care about but the exact value of which you don't know. Now I tell you that $x$ is not a prime greater than $42$.
Then you know that EITHER $x$ is not prime, OR $x$ is not greater than $42$ (or perhaps both!). This is an instance of the law $$\neg (A\wedge B)\quad\equiv\quad (\neg A)\vee (\neg B)$$ (here $A$ is "$x$ is a prime number" and $B$ is "$x$ is greater than $42$"). In general, we can use rules like this to break down complicated properties ("prime greater than $42$") in ways which may make them easier to reason about (e.g. we can try to prove things about $x$ by cases, handling the $\neg A$-case and the $\neg B$-case separately).