Computing With Logic Quiz (3)

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  1. Rewrite the following statements in formal way:

    • All real numbers have nonnegative squares. (Equivalent to Every Real number has a nonnegative square)

      $\forall X \in R, x^2 \geq 0$

    • All real numbers have squares not equal to -1. (Equivalent to No real numbers have squares equal to -1)

      $\forall X \in R, X^2 \text{ \}= -1$

    • There is a positive integer whose square is equal to itself. (Equivalent to Some positive integer equals its own square.)

      $\exist X \in Z^+ \text{ such that} X^2 = X$

  1. Consider the Tarski world, write the following statements using the formal logical notation:

    • There is a triangle x such that for all squares y, x is above y.

      $\exist X \text{ s.t. } \forall Y, triangle(X) \and (square(Y)\rightarrow above(X, Y))$

      $\exist X \text{ s.t. } triangle(X) \and (\forall Y, square(Y) \rightarrow above(X, Y))$

    • There is a triangle x such that for all circles y, x is above y.

      $\exist X, \text{ s.t. } triangle(X) \and (\forall Y, circle(Y)\rightarrow above(X, Y))$

      $\exist X \text{ s.t. } \forall Y, triangle(X) \and (circle(Y)\rightarrow above(X, Y))$

    • For all circles x, there is a square y such that y is to the right of x.

      $\forall X \text{ s.t. } circle(X)\rightarrow (\exist Y \text{ s.t. } square(Y) \and rightOf(Y, X))$

    • For all circles x and for all triangles y, x is to the right of y.

      $\forall X, \forall Y, (\text{ s.t. } circle(X) \and triangle(Y) \rightarrow rightOf(X, Y))$

    • There is a circle x and there is a square y such that x and y have the same color.

      $\exist X, \exist Y, \text{ s.t. } circle(X) \and triangle(Y) \and sameColor(X, Y)$

      $\exist X, \exist Y, \exist C, \text{ s.t. } circle(X) \and square(Y) \and color(X, C) \and color(Y, C)$

      $\exist X, \exist Y, \exist C, \text{ s.t. } circle(X) \and square(Y) \and C(X) \and C(Y) \and color(C)$

    • There is a circle x and there is a triangle y such that x and y have the same color.

      $\exist X, \exist Y, \text{ s.t. } circle(X) \and triangle(Y) \and sameColor(X, Y)$

  2. Let D = {-48, -14, -8, 0, 1, 3, 16, 23, 26, 32, 36}. Determine which of the following statements are true and which are false. Provide counterexamples for those statements that are false.

    • $\forall X \in D, \text{ if X is odd then } X > 0$

      True

    • $\forall X \in D, \text{ if X is less than 0 then X is even}$

      True

    • $\forall X \in D, \text{ if X is even then X < 0}$

      False, counterexample: for 16, true $\rightarrow$ false => false

    • $\forall X \in D, \text{ if the ones digit of X is 2, then the tens digit is 3 or 4}$

      True

    • $\forall X \in D, \text{ if the ones digit of X is 6, then the tens digit is 1 or 2}$

      False, counterexample: 36

  3. Write a negation for each statement:

    • $\forall X \in R, \text{ if } X^2 \geq 1 \text{ then } X > 0$

      $\exist X \in R, \text{ s.t. } X^2 \geq 1 \and X \leq 0$

    • $\forall D \in Z, \text{6/D is } \in Z \text{ then } D = 3$

      $\exist D \in Z, \text{ s.t. 6/D} is \in Z \and D \text{ \}= 3 $

    • $\forall X \in R, \text{ if } X(X+1) > 0 \text{ then } X > 0 \text{ or } X < -1$

      $\exist X \in R, \text{ s.t. } X(X+1)>0 \and (X\leq0 \and X\geq -1)$

    • $\forall X \in Z, \text{if X is prime then X is odd or X = 2}$

      $\exist X \in Z, \text{ s.t. X is prime} \and (X \text{ is even} \and X \text{ \}=2)$

    • $\forall A, B, C \in Z, \text{ if A - B is even and B - C is even then A - C is even}$

      $\exist A, B, C \in Z, \text{ s.t. } (\text{ A - B is even and B - C is even}) \and \text{ A - C is odd}$

    • $\forall X \in Z, \text{ if X is divisible by 6, then X is divisible by 2 and X is divisible by 3}$

      $\exist X \in Z, \text{ s.t. X is divisible by 6} \and (\text{ X is not divisible by 2 } \or \text{ X is not divisible by 3})$

    • If the square of an integer is odd, then the integer is odd.

      $\exist X \in Z, \text{ s.t. } X^2 \text{ is odd } \and \text{ X is even}$

    • If a function is differentiable then it is continuous

      $\exist \text{F a function, s.t. F is differentiable} \and \text{F is not continuous}$

  4. Write the negation for each statement:

    • $\forall \text{ colors C,}\exist \text{ an animal A s.t. A is colored C}$

      $\exist \text{ color C}, \forall \text{ animals A, s.t. A is not colored C}$

    • $\exist \text{ a book B s.t. } \forall \text{ people P, P has read B}$

      $\forall \text{ books B}, \exist \text{ a person P s.t. P has not read B}$

    • $\forall \text{ odd integers N, } \exist \text{ an integer K such that N = 2K + 1}$

      $\exist \text{ a odd integer N s.t. } \forall \text{ integers K, N \= 2K + 1}$

    • $\exist \text{ a real number U such that } \forall \text{ real numbers V, U*V = V}$

      $\forall \text{ real numbers U, } \exist \text{ a real number V s.t. U*V \= V}$

    • $\forall R \in Q, \exist \text{ integers A and B such that R = A/B}$

      $\exist R \in Q \text{ s.t. } \forall \text{integers A and B, R \= A/B}$

    • $\forall X \in R, \exist \text{ a real number Y such that X + Y= 0}$

      $\exist X \in R, \text{ s.t. } \forall \text{ real numbers Y, X + Y \= 0}$

    • $\exist X \in R \text{ such that for all real numbers Y, X + Y = 0}$

      $\forall X \in R, \exist \text{ a real number Y s.t. X + Y \= 0}$

  5. Use valid argument forms to show that the conclusion is a consequence of the premises (you should rewrite the statements in if-then logical form in Tarski’s World and replace some statements by their contrapositives).

    If an object is above all the triangles, then it is above all the blue objects.

    If an object is not above all the gray objects, then it is not a square.

    Every black object is a square.

    Every object that is above all the gray objects is above all the triangles.

    $\therefore$ If an object is black, then it is above all the blue objects.