Discrete Maths (HW/Q6)

Definitions of Variables and Domain

• Let P(x) denote the statement "x is a professional athlete".

• Let Q(x) denote the statement "x plays soccer".

• The domain of discourse is the set of all people in the world.

Proposition (i)

• Symbolic Form: $\forall x, (P(x) \land Q(x))$

• Description: "For all people x, x is a professional athlete and x plays soccer."

• Truth Value:

  • This statement is false because not all professional athletes play soccer.

• Negation:

  • Symbolically, negation is written as: $\neg(\forall x, (P(x) \land Q(x)))$

  • In words, this negation means: "There exists at least one person x such that x is a professional athlete and x does not play soccer."

Proposition (ii)

• Symbolic Form: $\exists x, (P(x) \lor Q(x))$

• Description: "There exists a person x such that x is a professional athlete or x plays soccer."

• Truth Value:

  • This statement is true because there are people who are either professional athletes, play soccer, or both.

• Negation:

  • Symbolically, negation is written as: $\neg(\exists x, (P(x) \lor Q(x)))$

  • In words, this negation means: "For all people x, x is neither a professional athlete nor does x play soccer."

Proposition (iii)

• Symbolic Form: $\forall x, (P(x) \to Q(x))$

• Description: "For all people x, if x is a professional athlete, then x plays soccer."

• Truth Value:

  • This statement is false because not all professional athletes play soccer (e.g., athletes in sports like basketball or football).

• Negation:

  • Symbolically, negation is written as: $\neg(\forall x, (P(x) \to Q(x)))$

  • In words, this negation means: "There exists at least one person x such that x is a professional athlete and x does not play soccer."

Proposition (iv)

• Symbolic Form: $\exists x, (P(x)Q(x))$

• Description: "There exists a person x such that x is a professional athlete and x plays soccer."

• Truth Value:

  • This statement is true since there are professional athletes who play soccer.

• Negation:

  • Symbolically, negation is written as: $\neg(\exists x, (P(x)Q(x)))$

  • In words, this negation means: "For all people x, if x is a professional athlete, then x does not play soccer."