$($Answer must be in your own hand$-$writing$)$

$($a$)$ Show that $\text{'}\text{'}$ and $\text{'}V\text{'}$ form a functionally complete collection of logical operators.

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$($b$)$ Let $T(x),F(x),$ and $B(x)$ be the statements $"x$ is a tiger," $"x$ is fierce," and $"x$ is beautiful," respectively. Express each of the following statements using quantifiers, logical connectives, and $T(x),F(x),$ and $B(x).$

$($i$)$ Tigers are fierce.

$($ii$)$ Some tigers are beautiful.

$($iii$)$ Some fierce things are beautiful.

Does $($iii$)$ follow from $($i$)$ and $($ii$)?$ Justify your answer.

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