in python $1.$ Write the following statements in symbolic form using the sym$-$

bols $,,$ and $$ and the indicated letters to represent component

statements.

Let h represent the statement $$John is healthy,$$ w represent the statement

$$John is wealthy,$$ and s represent the statement $$John is wise.$$

$($a$)$ John is healthy and wealthy but not wise.

$($b$)$ John is not wealthy but he is healthy and wise.

$($c$)$ John is neither healthy, wealthy, nor wise.

$($d$)$ John is neither wealthy nor wise, but he is healthy.

$($e$)$ John is wealthy, but he is not both healthy and wise.

$2.$ Write truth tables for the statement forms

$($a$)($p $$ q$)($p $$ q$)$

$($b$)$ p $($ q $$ r$)$

$3.$ Determine whether the statement forms are logically equivalent.

In each case, construct a truth table and include a sentence

justifying your answer. Your sentence should show that you

understand the meaning of logical equivalence.

$1$

$($a$)$ p $($p $$ q$)$ and q

$($b$)($p $$ q$)$ and $$ p$$ q

$($c$)$ p $($q $$ r$)$ and $($p $$ q$)($p $$ r$)$

$4.$ Assume x is a particular real number and use De Morgan$$s laws

to write negations for the statements

$($a$)2<$ x $<6$

$($b$)9<$ x $<2$

$($c$)$ x $<2$ or x $>6$

$($d$)$ x $<=1$ or x $>1$

$($e$)0>$ x $>=4$

$5.$ Use truth tables to establish which of the statement forms are

tautologies and which are contradictions.

$($a$)($p $$ q$)($ p $($p$$ q$))$

$($b$)($p$$ q$)($ p $$ q$)$

$($c$)(($ p $$ q$)($q $$ r$))$ q

$6.$ Determine whether the statements in $($a$)$ and $($b$)$ are logically

equivalent.

$($a$)$ Bob is a double math and computer science major and Ann is a math

major, but Ann is not a double math and computer science major.

$($b$)$ It is not the case that both Bob and Ann are double math and com$-$

puter science majors, but it is the case that Ann is a math major

and Bob is a double math and computer science major.

$7.$ Using Exclusive OR

Is $($p $$ q$)$ r $$ p $($q $$ r$)?$ Justify your answer.

$8.$ What logical equivalence is used for each step.

$($p$$ q$)($p $$ q$)$ p $($ q $$ q$)$ by $($a$)$

$$ p $($q$$ q$)$ by $($b$)$

$$ p $$ t by $($c$)$

$$ p by $($d$)$

Therefore, $($p$$ q$)($p $$ q$)$ p$.$

Verify the logical equivalences. Supply a reason for each step.

$9.($p$$ q$)$ p $$ p

$10.(($ p $$ q$)($ p$$ q$))($p $$ q$)$ p