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Q)Consider these statements, of which the first three are and fourth is a valid conclusion.

"All hummingbirds are richly colored."

"No large birds live on honey."

"Birds that do not live on honey are dull in color"

"Hummingbirds are small."

Express using quantifiers??
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P(x)=x is richly colored.

Q(x)=x is large.

R(x)=x lives on honey.

T(x)=x is a hummingbird.

What are we given?

In English- All hummingbirds are richly colored and No large birds live on honey and Birds that do not live on honey are dull in color.

First, write given three propositions-

"All hummingbirds are richly colored."

 ∀x(T(x)->P(x))

"No large birds live on honey."

This can be simplified as- All large birds do not live on honey.

∀x(Q(x)->~R(x)) or ∀x(R(x)->~Q(x))   [p->q =~q->~p]

"Birds that do not live on honey are dull in color"

∀x(~R(x)->~P(x))  or ∀x(P(x)->R(x))  [p->q =~q->~p]

Now prove Conclusion-

In English, it should be All humming birds are small.

∀x(T(x)->P(x)) AND ∀x(R(x)->~Q(x)) AND ∀x(P(x)->R(x)) 

∀x(T(x)->P(x) AND P(x)->R(x) AND R(x)->~Q(x))

∀x(T(x)->P(x) AND P(x)->R(x) AND R(x)->~Q(x)) -> ∀x(T(x)->~Q(x))   [Why? because of Law of syllogism]

So, with this we come to conclusion - ∀x(T(x)->~Q(x)) which means If x is a humming bird then it is not large or All Humming birds are small.

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