Question 1: If A, B, and C is true statements and X, Y, and Z, are false statements, determine the truth value of the following:

a) [Z ⋁ (Y · X)] ⋁ ~ [(X ⋁ Y) · (X ⋁ Z)]

[F ⋁ (F · F)] ⋁ ~ [(F ⋁ F) · (F ⋁ F)]

[F ⋁ (F · F)] ⋁ ~ [F · F]

[F ⋁ F] ⋁ ~ F

F ⋁ ~F

F ⋁ T

T

b) [C · (B ⋁ A)] · ~ [(A · B) ⋁ (A · C)]

[T · (T ⋁ T)] · ~ [(T · T) ⋁ (T · T)]

[T · (T ⋁ T)] · ~ [T ⋁ T]

[T · T] · ~ T

T · F

F

c) ~ {[(~A · C) · (~X · Z)] · ~[(A · ~C) ⋁ ~(~Y · ~Z)]}

~ {[(~T · T) · (~F · F)] · ~[(T · ~T) ⋁ ~(~F · ~F)]}

~{[(F · T) · (T · F)] · ~[(T · F) ⋁ ~(T · T)]}

~{[F · F] · ~[ F ⋁ ~T ]}

~{[F · F] · ~[ F ⋁ F ]}

~ {F · ~F}

~ {F · T}

~F

T

d) {[(Y ⊃ X) ⊃ Z] ⊃ [Z ⊃ (X ⊃ Y)]} ⊃ [(X ⊃ Z) ⊃ Y]

{[(F ⊃ F) ⊃ F] ⊃ [F ⊃ (F ⊃ F)]} ⊃ [(F ⊃ F) ⊃ F]

{[T ⊃ F] ⊃ [F ⊃ T]} ⊃ [T ⊃ F]

{F ⊃ T} ⊃ F

T ⊃ F

F

e) [(A · Z) ⊃ Y] ⊃ [(A ⊃ Z) · (A ⊃ Y)]

[(T · F) ⊃ F] ⊃ [(T ⊃ F) · (T ⊃ F)]

[F ⊃ F] ⊃ [F · F]

T ⊃ F

F

Question 2

Use truth tables to determine the validity or invalidity of each of the following arguments:

a) (R ⋁ S) ⊃ T

T ⊃ (R · S)

Therefore, (R · S) ⊃ (R ⋁ S)

Solution

Components | Premises | Conclusion | |||

R | S | T | (R ⋁ S) ⊃ T | T ⊃ (R · S) | (R · S) ⊃ (R ⋁ S) |

T | T | T | T | T | T |

T | T | F | F | T | T |

T | F | T | T | F | T |

T | F | F | F | T | T |

F | T | T | T | F | T |

F | T | F | F | T | T |

F | F | T | T | F | T |

F | F | F | T | T | T |

The argument is valid. In the First and last line, both premises are true and the conclusion is also true. So the argument is considered to be true.

b) If coal consumption continues to grow, then either coal imports will increase or domestic coal reserves will be depleted.

If coal imports increase and domestic coal reserves are depleted, then Australia will eventually go bankrupt.

Therefore, if coal consumption continues to grow, then Australia will eventually go bankrupt.

Solution:

Let P be the statement: coal consumption continues to grow

Let Q be the statement: coal imports will increase

Let R be the statement: domestic coal reserves will be depleted

Let S be the statement: Australia will eventually go bankrupt..............

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