Limpet - Testing document 1

Example 1

Formulate the compound proposition C by means of simple propositions and logical operators.

Proposition A: I will go to do shopping.

Proposition B: I will prepare a cup of tea.

Proposition C: $A\Leftrightarrow \neg B$

Answer:

Example 2

Is the formula $\alpha =(\neg X\vee \neg Y)\Rightarrow (Z\Rightarrow Z)$ satisfiable?

Is the formula $\beta =(\neg X\vee \neg X)\wedge \neg (Y\Rightarrow Z)$ a tautology?

Are the following two formulas $\varphi_1, \varphi_2$ logically equivalent?
$\varphi_1 =(\neg X\wedge \neg Y) \vee \neg (Z\wedge Z)$
$\varphi_2 =(X\vee \neg Y)\vee \neg (Z\Rightarrow Z)$

Example 3

Find negations of the following formulas:

$\Xi =\exists x[(P(x)\wedge Q(x))\vee R(x)]$

$\Pi =\forall x[P(x)\Rightarrow \forall yQ(y)]$

$\Omega =\forall x[P(x)\Rightarrow Q(x)]\wedge \exists x[R(x)\wedge S(x)]$

Example 4

The following sets are given: $A=\{0,4,5\},\ B=\{0,2,3,4\},\ C=\{3,4\},\ D=\{2,4,5\}$ . The set $\Theta$ is determined by the following set operations:

$\Theta =[(D\setminus C)\times (B\cap A)]\setminus [(A\cup B)\times C]$

Write down all the elements of the set $\Theta$ and specify their quantity. How many elements does the power set of the set $\Theta$ contain? (You don't need to write the power set's elements)

Example 5

a) $A\cap B=\emptyset$

b) $A=B$

c) $A\subseteq B$

d) $A\subset B$

e) $A\ne B$

Example 6

Solve the following inequations in $\mathbb{R}$ :

$|6x^2-5x|<6$

$|x-2|\ge 3\wedge |x+1|\le 7$

Example 7

Solve the following equation in $\mathbb{R}.$ 2^{-x^2 +x +2} =4^{ x^2 +x +1}$.