Question

Let p and q be the propositions “The election is decided” and “The votes have been counted,” respectively. Express each of these compound propositions as an English sentence.

a)$\mathbf{\text{¬p}}$

b)$\mathbf{\text{p∨q}}$

c)$\mathbf{\text{¬p∧q}}$

d)$\mathbf{\text{q→p}}$

e)$\mathbf{\text{¬q→¬p}}$

f )$\mathbf{\text{¬p→¬q}}$

g) $\mathbf{\text{p↔q}}$

h) $\mathbf{\text{¬q∨(¬p∧q)}}$

Short answer

1. The election is not decided.
2. The election is decided or the votes have been counted.
3. The election is not decided and the votes have been counted.
4. If the votes have been counted then the election is decided.
5. If the votes have not been counted then the election is not decided.
6. If the election is not decided then the votes have not been counted.
7. The election is decided if and only if the votes have been counted.
8. The votes have not been counted or the election is not decided and the votes have been counted.

Step-by-step solution

Definition of proposition

Proposition:
1. It is a declarative statement which can be either true or false.
2. It cannot be both true and false simultaneously.

English sentence for a)

q is “The votes have been counted”

$\mathbf{\text{¬p}}$means,

“The election is not decided”

Step 3:English sentence for b)

We write given statement as,

$\mathrm{p}\vee \mathrm{q}$would mean,

The election is decided or the votes have been counted.

English sentence for c)

We write given statement as,

$\neg \mathrm{p}\wedge \mathrm{q}$would mean,

The election is not decided and the votes have been counted.

English sentence in d)

We write the given statement as,

$\mathrm{q}\to \mathrm{p}$would mean,

If the votes have been counted then the election is decided.

English sentence in e)

We write given statement as,

$\neg \mathrm{q}\to \neg \mathrm{p}$would mean,

If the votes have not been counted then the election is not decided.

English sentence in f)

We write given statement as,

$\text{¬p→¬q}$would mean,

If the election is not decided then the votes have not been counted.

English sentence in g)

We write given statement as,

$\text{p↔q}$would mean,

The election is decided if and only if the votes have been counted.

English sentence in h)

We write given statement as,

$\text{¬q∨(¬p∧q)}$would mean,

The votes have not been counted or the election is not decided and thevotes have been counted.