Question

Suppose that the domain of Q(x,y,z) consists of triples x, y, z, where x = 0, 1 or 2, y = 0, or 1and z = 0 or 1. Write out these propositions using disjunctions and conjunctions.

$$\begin{gathered} \mathbf{a}\mathbf{)}\mathbf{}\mathbf{\forall }\mathbf{yQ}\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{y}\mathbf{,}\mathbf{0}\mathbf{\left)} \\ \mathbf{b}\mathbf{\right)}\mathbf{}\exists \mathbf{xQ}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{1}\mathbf{)} \\ \mathbf{C}\mathbf{)}\mathbf{}\exists \mathbf{z}\mathbf{\neg }\mathbf{Q}\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{z}\mathbf{\left)} \\ \mathbf{d}\mathbf{\right)}\mathbf{}\exists \mathbf{x}\mathbf{\neg }\mathbf{Q}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{)} \end{gathered}$$

Short answer

a) The exists $\mathbf{\forall }\mathbf{yQ}\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{y}\mathbf{,}\mathbf{0}\mathbf{)}$propositions using disjunctions and conjunctions can be expressed as$\mathbf{Q}\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{)}\mathbf{\wedge }\mathbf{Q}\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{0}\mathbf{)}$

b) The$\exists \mathbf{xQ}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{1}\mathbf{x}\mathbf{1}\mathbf{)}$propositions using disjunctions and conjunctions can be express as$\mathbf{Q}\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{1}\mathbf{)}\mathbf{\vee }\mathbf{Q}\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{1}\mathbf{)}\mathbf{\vee }\mathbf{Q}\mathbf{(}\mathbf{2}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{1}\mathbf{)}$

c) The exists$\exists \mathbf{z}\mathbf{\neg }\mathbf{Q}\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{z}\mathbf{)}$propositions using disjunctions and conjunctions can be express as$\mathbf{\neg }\mathbf{Q}\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{)}\mathbf{\vee }\mathbf{\neg }\mathbf{Q}\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{)}$

d) The $\exists \mathbf{x}\mathbf{\neg }\mathbf{Q}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{)}$propositions using disjunctions and conjunctions can be express as$\mathbf{\neg }\mathbf{Q}\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{)}\mathbf{\vee }\mathbf{\neg }\mathbf{Q}\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{)}\mathbf{\vee }\mathbf{\neg }\mathbf{Q}\mathbf{(}\mathbf{2}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{)}$

Step-by-step solution

Q(0,0,0)∧Q(0,1,0)

ycan taken on values 0 and 1

Q(0,1,1)∨Q(1,1,1)∨Q(2,1,1)

xcan take on values 0,1 and 2.

¬Q(0,0,0)∨¬Q(0,0,1)

zcan take on values 0and 1.

¬Q(0,0,1)∨¬Q(1,0,1)∨¬Q(2,0,1)

x can take on values 0,1 and2.