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thedeathlord123
21.04.2021 •
Mathematics
The proof that is shown.
Given: ΔMNQ is isosceles with base , and and bisect each other at S.
Prove:
Square M N Q R is shown with point S in the middle. Lines are drawn from each point of the square to point S to form 4 triangles.
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Ответ:
(1) Contradiction
(2) Tautology
(3) Neither Tautology nor Contradiction
(4) Tautology
Step-by-step explanation:
Given from the option that ,
(1) (P ↔ Q) ↔ ((P ∧ ¬Q) ∨ (¬P ∧ Q))
(2) (P → Q) V (Q →P)
(3) (¬R→((P V Q) → Q)) V R
(4) ((P→Q) ∧ (Q→R)) → (P→R)
From the first option (1), we have that;
(1)(P ↔ Q) ↔ ((P ∧ ¬Q) ∨ (¬P ∧ Q))
PQ¬P¬QP ↔ QP ∧ ¬Q¬P ∧Q(P∧¬Q) V (¬P∧Q) result
TTFTFFFFF
TFFTFTFTF
FTTFFFTTF
FFTTTFFFF
From the result in the last column, this is a contradiction.
From option (2) we have that;
(2)(P → Q) V (Q →P)
The resulting table becomes
PQP →QQ → P result
TTTTT
TFFTT
FTTFT
FFTTT
It is seen from the table that the result of the last column has all T which implies that it is a Tautology.
From option (3) we have that;
(3) (¬R→((P V Q) → Q)) V R
The resulting table becomes
P QR¬RPVQ(PVQ)→Q¬R→(PVQ→Q)Result
TTTFTTTT
TTFTTTTT
TFTFTFTT
TFFTTFFF
FTTFTTTT
FTFTTTTT
FFTFFTTT
FFFTFTTT
There is a little difference in the last column when compared to others,
Here we can see that the last column contains all T but just one F, this implies that it is neither a Tautology nor Contradiction.
From option (4) we have that;
(4) ((P→Q) ∧ (Q→R)) → (P→R)
The resulting table becomes
PQRP →QQ →R(P→Q)∧(Q→R)P→R result
TTT T T T T T
TTF T F F F T
TFT F T F T T
TFF F T F F T
FTT T T T T T
FTF T F F T T
FFT T T T T T
FFF T T T T T
Here we can see that the last column contains all T which implies that it is a Tautology.
i hope this helps, cheers.