Question

for each pair of sets, choose the best description. pair of sets description (a) $a = \\{81, 82, 83\\}$ $b = \\{53, 54, 55\\}$ $\bigcirc$ equivalent but not equal $\bigcirc$ equal but not equivalent $\bigcirc$ both equivalent and equal $\bigcirc$ neither equivalent nor equal (b) $a$ is the set of even numbers greater than 11 and less than 19. $b = \\{12, 14, 16, 18\\}$ $\bigcirc$ equivalent but not equal $\bigcirc$ equal but not equivalent $\bigcirc$ both equivalent and equal $\bigcirc$ neither equivalent nor equal (c) $a$ is the set of integers greater than 16 and less than 20. $b$ is the set of integers greater than 16. $\bigcirc$ equivalent but not equal $\bigcirc$ equal but not equivalent $\bigcirc$ both equivalent and equal $\bigcirc$ neither equivalent nor equal (d) $a = \\{h, j, m, p\\}$ $b = \\{p, d, g\\}$ $\bigcirc$ equivalent but not equal $\bigcirc$ equal but not equivalent $\bigcirc$ both equivalent and equal $\bigcirc$ neither equivalent nor equal

Explanation:

Step1: Define key terms

Two sets are equal if they have exactly the same elements. Two sets are equivalent if they have the same number of elements (same cardinality).

Step2: Solve part (a)

Count elements: $n(A)=3$, $n(B)=3$ (equivalent). Elements differ, so not equal.

Step3: Solve part (b)

Set $A$ = {12,14,16,18}, which matches $B$ exactly. So they are equal (and thus equivalent).

Step4: Solve part (c)

Set $A$ = {17,18,19} ($n(A)=3$). Set $B$ has infinitely many elements. Different cardinality, so neither equivalent nor equal.

Step5: Solve part (d)

Count elements: $n(A)=4$, $n(B)=3$. Different cardinality, so neither equivalent nor equal.

Answer:

(a) equivalent but not equal
(b) both equivalent and equal
(c) neither equivalent nor equal
(d) neither equivalent nor equal