Question

Consider statements p, q, and r. p: It is raining. q: Sam is wearing a hat. r: It is winter. For each part below, fill in the symbolic form. Descriptive form Symbolic form (a)If it is raining then Sam is not wearing a hat, and it is not winter. (b)It is winter or Sam is wearing a hat, if and only if it is not raining.

Answer 1

a)
`p\rightarrow \\eg q \wedge \\eg r`

b)
`r\vee q \iff \\eg p`

Explanation:

Given statements:

`p:` It is raining

`q:` Sam is wearing a hat

`r:` It is winter

To find:

The symbolic forms for the following:

a) If it is raining then Sam is not wearing a hat, and it is not winter.

b) It is winter or Sam is wearing a hat, if and only if it is not raining.

Solution:

a) The statement followed by 'if ' is written on the left hand side and the statement followed by 'then' is written after the symbol
`\rightarrow .`

Not of a statement
`s` is represented as
`\\eg s`.

And is represented by
`\wedge` symbol.

Therefore, the statement in symbolic form becomes:

`p\rightarrow \\eg q \wedge \\eg r`

b) Not of a statement
`s` is represented as
`\\eg s`.

Or is represented by
`\vee` symbol.

If and only if is represented by the symbol
`\iff.`

Therefore, the statement in the symbolic form becomes:

`r\vee q \iff \\eg p`