Day: October 22, 2020

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202010220700 Problem 1, Chapter 1.1

The Fahrenheit temperature scale is defined so that ice melts at 32^\circ \textrm{F} and water boils at 212^\circ \textrm{F}.

(a) Derive the formulas for converting from Fahrenheit to Celsius and back.

(b) What is absolute zero on the Fahrenheit scale?

Solution.

Given that the melting point is 32^\circ\textrm{F} and the boiling point 212^\circ \textrm{F}.

Therefore 1\,\textrm{C}^\circ =\displaystyle{\frac{212-32}{100}}\textrm{F}^\circ = 1.8\,\textrm{F}^\circ.

Then F=32+1.8C and C=\displaystyle{\frac{F-32}{1.8}}.

Absolute zero on Fahrenheit scale is F=32+1.8(-273)=-459.4^\circ\textrm{F}.

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202010220448 Exercises 2.1A (Q1)

Suppose |S|=19, |T|=11 and |S\cap T|=8. Find |S\cup T| and |S\backslash T|.

Hint. (Verbal translation)

You are given that the number of elements in set S is 19,the number of elements in set T is 11, and the number of elements in the intersection of set S and set T is 8.

You are asked:

What is the number of elements in the union of set S and set T?What is the number of elements in the relative complement S\backslash T of set T with respect to set S?

Definition. The relative complement of T with respect to S is the set

S\backslash T=\{ x\, |\enspace x\in S\textrm{ and }x\notin T\}.

Can you try drawing a Venn diagram?

Attempts.

(constructive)

Let

S=\{ a,\, b,\, c,\, d,\, e,\, f,\, g,\, h,\, i,\, j,\, k,\, l,\, m,\, n,\, o,\, p,\, q,\, r,\, s\},

and also

T=\{ l,\, m,\, n,\, o,\, p,\, q,\, r,\, s,\, t,\, u,\, v\},

so that S\cap T=\{ l,\, m,\, n,\, o,\, p,\, q,\, r,\, s \}.

The union S\cup T of set S and set T must as follows be:

S\cup T=\{ a,\, b,\, c,\, d,\, e,\, f,\, g,\, h,\, i,\, j,\, k,\, l,\, m,\, n,\, o,\, p,\, q,\, r,\, s,\, t,\, u,\, v\},

such that |S\cup T|=22.

The relative complement of set T w.r.t. set S is

S\backslash T=\{ a,\, b,\, c,\, d,\, e,\, f,\, g,\, h,\, i,\, j,\, k\}

and the number of its elements is

|S\backslash T| = 11.

(analytic)

By observation of the Venn diagram,

you are writing out

|S\cup T|=|S|+|T|-|S\cap T|,

keeping in mind that

|S\cap T|=|S|+|T|-|S\cup T|

shall answer another question of a different subject.