Question

for each pair of sets, choose the best description.
(a) $a = {81, 82, 83}$, $b={52, 53, 54}$
description options: equivalent but not equal, equal but not equivalent, both equivalent and equal, neither equivalent nor equal
(b) $a$ is the set of even numbers greater than 7 and less than 15. $b = {8, 10, 12, 14}$
description options: equivalent but not equal, equal but not equivalent, both equivalent and equal, neither equivalent nor equal
(c) $a$ is the set of integers greater than 6 and less than 10. $b$ is the set of integers greater than 6.
description options: equivalent but not equal, equal but not equivalent, both equivalent and equal, neither equivalent nor equal
(d) $a={h,j,m,p}$, $b = {m,d,c}$
description options: equivalent but not equal, equal but not equivalent, both equivalent and equal, neither equivalent nor equal
question 5 of 30
here are the meanings of some of the symbols that appear in the statements below.

• $subseteq$ means \is a subset of.\
• $subset$ means \is a proper subset of.\
• $

subseteq$ means \is not a subset of.\

• $varnothing$ is the empty set.

for each statement, decide if it is true or false.
(a) ${u, v}subsetvarnothing$
true false
(b) ${1, 2, 3, 4}
subseteq{1, 3}$
true false
(c) ${c,f,h}subset{c,d,f,g,h,j}$
true false
(d) ${11, 14, 15}subseteq{11, 14, 15}$
true false

Explanation:

Step1: Recall set - equivalence and equality

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

Step2: Analyze pair (a)

Set \(A=\{81,82,83\}\) and \(B = \{52,53,54\}\). Both sets have 3 elements, so they are equivalent. But the elements are different, so they are not equal.

Step3: Analyze pair (b)

Set \(A\) of even numbers greater than 7 and less than 15 is \(A=\{8,10,12,14\}\), and \(B = \{8,10,12,14\}\). They have the same elements, so they are both equal and equivalent.

Step4: Analyze pair (c)

Set \(A=\{7,8,9\}\) and \(B\) is the set of integers greater than 6 (\(B=\{7,8,9,\cdots\}\)). \(A\) has 3 elements and \(B\) is an infinite - set. They are neither equal nor equivalent.

Step5: Analyze pair (d)

Set \(A=\{h,j,m,p\}\) has 4 elements and \(B=\{m,d,c\}\) has 3 elements. They are neither equal nor equivalent.

Step6: Recall subset definitions

\(\subseteq\) means all elements of the left - hand set are in the right - hand set. \(\subset\) means all elements of the left - hand set are in the right - hand set and the right - hand set has more elements. \(
subseteq\) means there is at least one element in the left - hand set that is not in the right - hand set.

Step7: Analyze statement (a)

The empty set \(\varnothing\) has no elements. So, \(\{u,v\}\) cannot be a proper subset of \(\varnothing\). The statement \(\{u,v\}\subset\varnothing\) is false.

Step8: Analyze statement (b)

The set \(\{1,2,3,4\}\) has elements 2 and 4 which are not in \(\{1,3\}\). So, \(\{1,2,3,4\}
subseteq\{1,3\}\) is true.

Step9: Analyze statement (c)

All elements of \(\{c,f,h\}\) are in \(\{c,d,f,g,h,j\}\), and \(\{c,d,f,g,h,j\}\) has more elements. So, \(\{c,f,h\}\subset\{c,d,f,g,h,j\}\) is true.

Step10: Analyze statement (d)

A set is always a subset of itself. So, \(\{11,14,15\}\subseteq\{11,14,15\}\) is true.

Answer:

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