A B C KB S1 True True True True True True True False False True True.

Question

Question

A B C KB S1 True True True True True True True False False True True.

A	B	C	KB	S1
True	True	True	True	True
True	True	False	False	True
True	False	True	True	True
True	False	False	False	True
False	True	True	False	False
False	True	False	False	False
False	False	True	True	True
False	False	False	False	False

KB and S1 are two propositional logic statements, that are constructed using symbols A, B, C, and using various connectives. The above truth table shows, for each combination of values of A, B, C, whether KB and S1 are true or false.

Part a: Given the above information, does KB entail S1? Justify your answer.

Part b: Given the above information, does statement NOT(KB) entail statement NOT(S1)? Justify your answer.

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Answer 1

Answer #1

Solution:

Given,

A	B	C	KB	S1
True	True	True	True	True
True	True	False	False	True
True	False	True	True	True
True	False	False	False	True
False	True	True	False	False
False	True	False	False	False
False	False	True	True	True
False	False	False	False	False

(a)

Explanation:

=>KB entail S1 means KB => S1

Checking KB => S1:

=>KB => S1 = ~KB v S1 where v means OR operator

A	B	C	KB	S1	KB => S1
True	True	True	True	True	True
True	True	False	False	True	True
True	False	True	True	True	True
True	False	False	False	True	True
False	True	True	False	False	True
False	True	False	False	False	True
False	False	True	True	True	True
False	False	False	False	False	True

=>As KB => S contains all the values "True" so we can say that KB entail S1.

(b)

Explanation:

=>NOT(KB) entail NOT(S1) means ~ KB => ~ S1 where ~ means negation operator

Checking ~ KB => ~ S1 :

=>~ KB => ~ S1 = ~(~ KB) v ~ S1

=>~ KB => ~ S1 = KB v ~S1 as ~( ~ KB) = KB

A	B	C	KB	S1	~ KB => ~S1
True	True	True	True	True	True
True	True	False	False	True	False
True	False	True	True	True	True
True	False	False	False	True	False
False	True	True	False	False	True
False	True	False	False	False	True
False	False	True	True	True	True
False	False	False	False	False	True

=>As some entries(truth values) of ~ KB => ~ S1are false so NOT(KB) does not entail NOT(S1).

I have explained each and every part with the help of statements attached to it.

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Answer 2

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A B C KB S1 True True True True True True True False False True True.

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