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

false

step-by-step explanation:

whenever we assume a point on a plane then according to geometry it will definitely have 2 dimensions and if it assumed in space then it will have 3 dimensions.

Answer from: Quest

a) one student is chosen from classroom a, then a student is chosen from classroom b.

step-by-step explanation:

independent events are ones in which the probability of one event does not affect the probability of the second event.

one student is chosen from classroom a; this does not affect the chances of choosing a student from classroom b. this makes these independent events.

when one card is chosen from a standard deck and then set aside before a second card is chosen, the probability of the second card being chosen changes. this makes them dependent events.

when one student is chosen from classroom a and then a second student is chosen from classroom a, the probability of choosing the second student changes. this makes them dependent events.

when a letter is chosen from the alphabet and then a second letter is chosen, the probability of choosing the second letter changes. this makes them dependent events.

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Prove that for all sets a and b, if p(a) ⊆ p(b), then a ⊆ b (we can take generic sets a and b, assum...

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