Question 1209177: Premise:
H & ( C & T )
~ ( ~ F & T )
Conclusion:
F
Answer by math_tutor2020(3816)   (Show Source):
You can put this solution on YOUR website!
I'll assume that these particular F and T symbols do not represent "false" and "true", but rather just any logical statement. Your professor should have chosen different letters.
Here's a direct derivation
| Number | Statement | Line(s) Used | Reason | | 1 | H & (C & T) | | | | 2 | ~( ~F & T ) | | | | :. | F | | | | 3 | (H & C) & T | 1 | Association | | 4 | T & (H & C) | 3 | Commutation | | 5 | T | 4 | Simplification | | 6 | ~(~T) | 5 | Double Negation | | 7 | ~(~F) v ~T | 2 | De Morgan’s Law | | 8 | F v ~T | 7 | Double Negation | | 9 | ~T v F | 8 | Commutation | | 10 | F | 9, 6 | Disjunctive Syllogism |
Here's a list of rules of inference and replacement
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Another way to do the derivation is to use an indirect proof (aka proof by contradiction)
| Number | Statement | Line(s) Used | Reason | | 1 | | H & (C & T) | | | | 2 | | ~( ~F & T ) | | | | :. | | F | | | | 3 | ~F | | Assumption for Indirect Proof | | 4 | ~(~F) v ~T | 2 | De Morgan’s Law | | 5 | F v ~T | 4 | Double Negation | | 6 | ~T | 5, 3 | Disjunctive Syllogism | | 7 | (H & C) & T | 1 | Association | | 8 | T & (H & C) | 7 | Commutation | | 9 | T | 8 | Simplification | | 10 | T & (~T) | 9, 6 | Conjunction | | 11 | | F | 3 - 10 | Indirect Proof |
Line 3 is where we assume the opposite of the conclusion we want to arrive at.
From there a chain event of dominoes fall over to lead to T & (~T) which is a contradiction. One of T or ~T is false, while the other is true. This contradiction means our assumption must be the opposite.
The assumption ~F led to a contradiction, which means the opposite (F) must be a valid conclusion.
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