Mathematics, 28.07.2019 10:00 kiwipup16

Prove that for all sets a and b, if p(a) ⊆ p(b), then a ⊆ b (we can take generic sets a and b, assume p(a) ⊆ p(b), and prove a ⊆ b. what is the typical technique to prove a set is a subset of another

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recall that the power set of a set a, denoted $\mathcal{p}(a)$ , is the set of all subsets of a. the subset includes the set a itself.

assume that $\mathcal{p}(a) \subseteq \mathcal{p}(b)$ for some generic sets a and b. symbolically, means that $\forall x(x \in \mathcal{p}(a) \rightarrow x \in \mathcal{p}(b) )$ (for all elements x from the universe, if x belongs to set $\mathcal{p}(a)$ , then x belongs to set $\mathcal{p}(b)$ ). now we have to show that $a \subseteq b$ .

since a is a subset of a, $a \in \mathcal{p}(a)$ . since we assumed that $\mathcal{p}(a) \subseteq \mathcal{p}(b)$ is true, it follows from $a \in \mathcal{p}(a)$ that $a \in \mathcal{p}(b)$ . by definition of power set, $a \subseteq b$ .

the typical technique to prove a set is a subset of another is true is to use a direct proof. the statement $x \subseteq y$ means that, for all elements x from the universe, $x \in x$ implies that $x \in y$ . thus we do a direct proof by

taking any arbitrary x from the universe,supposing that $x \in x$ is true (we do not consider the case where $x \notin x$ because a false premise implying anything is always a true implication), andthen use $x \in x$ to infer that $x \in y$ .

after we have made the inference, since we showed that $x \in y$ is true for any arbitrary, generic value x from the universe, the rule of universal generalization can be invoked to generalize that x is a subset of y.

Answer from: Quest

I’m a smart person so that’s why i think it was 6

Answer from: Quest

step-by-step explanation: so we look at the formula un = a + (n-1)d

un is the amount of seats

n is the nth term aka row number

a is the original sequence/row number

d is the difference between each term

so in this a=22 d=6 and n can remain as n because we are simply formjng a rule

lets put these in the equation

un=22+(n-1)(6)

this is your answer to part a

however

lets double check that this is correct, we'll try putting it into the second term

un=a+(n-1)d

u2=22+(2-1)(6)

u2=22+6

u2=28

as the question said the second row has 28 seats so now we know the formula we made is correct.

as for part b

un=a+(n-1)

its asking us to find a row with 100 seats.

remember un was the amount of seata so all we do is substitute this in.

un=100 a=22 d=6

un=a+(n-1)d

100=22+(n-1)(6)

so now we simplify to find n which is the row number.

100-22=22+(n-1)(6)-22

78=(n-1)(6)

78/6=(n-1)(6)/6

13=n-1

13+1=n-1+1

14=n

so now we know that in the 14th row there is 100 seats.

hope this and that you understood! its my first time using the app or anyone out on here : p its also me to revise so a bunch! ask more questions if you dont understand ☺

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