$($a$)$ Given the following propositions:

$d$ : it is day

$n$ : it is night

s: the sun shines

$m$ : the moon shines

Give propositions that respectively resemble the meaning of the following two sentences:

i$.$ The sun only shines when it is day, but the sun doesn't shine now although it is day.

ii$.$ Either it is day, or it is night, but when the sun shines it is not night.

$6$

$($b$)$ Indicate which of the following statements are true and which are false. Justify your answer.

i$.$ AAxinZ,EEyinR$-\{0\}$ such that $xy<mn\; id="EditorElement\_372670">1</mn>$

ii$.$ AAxinR$-\{0\},$AAyinZ such that $xy<mn\; id="EditorElement\_372579">1</mn>$

iii. AAxinR,EEyinR such that $xy=1$

$6$

$($c$)$ Let $p,q,r$ and $s$ be three propositions: Say whether or not the following argument is a valid argument. Explain your answer.

$sr$

$(pvvq)$notr

$(nots)(notqr)b$

$p$

:$.q$