Exercise 2.3

1. Which of the following sets are equal?

A = {x : x is a letter in the word FOLLOW}, B = {x : x is a letter in the word FLOW}

and C = {x : x is a letter in the word WOLF}

Solution: Set A: Letters in "FOLLOW"

Distinct letters: , , , (Enter in alphabetical order.)

Therefore, A = { , , , }

Set B: Letters in "FLOW"

Distinct letters: , , , (Enter in order of appearance)

Therefore, B = { , , , } (Enter in order of appearance)

Set C: Letters in "WOLF"

Distinct letters: , , ,

Therefore, C = { , , , }

Comparison: All three sets contain elements.

So, A = B = C. All three sets are .

2. Consider the following sets and fill up the blanks with = or ≠ so as to make the statement true.

A = {1, 2, 3}; B = {The first three natural numbers}

C = {a, b, c, d}; D = {d, c, a, b}

E = {a, e, i, o, u}; F = {set of vowels in English Alphabet}

Let's first write each set in roster form: A = {1, 2, 3} ; B = {The first three natural numbers} = { , , } ; C = {a, b, c, d}

D = {d, c, a, b} ; E = {a, e, i, o, u} ; F = {set of vowels in English Alphabet} = { , , , , }

(i) A .... B : A = {1, 2, 3} and B = {1, 2, 3}: Since both sets have elements: A B

(ii) A .... E : A = {1, 2, 3} and E = {a, e, i, o, u}. These sets have elements: A E

(iii) C .... D : C = {a, b, c, d} and D = {d, c, a, b}. Since both sets contain elements: C D

(iv) D .... F : D = {a, b, c, d} and F = {a, e, i, o, u}. These sets have elements: D F

(v) F .... A : F = {a, e, i, o, u} and A = {1, 2, 3}. These sets have elements: F A

(vi) D .... E : D = {a, b, c, d} and E = {a, e, i, o, u}. These sets have elements: D E

(vii) F .... B : F = {a, e, i, o, u} and B = {1, 2, 3}. These sets have elements: F B

3. In each of the following, state whether A = B or not.

(i) A = {a, b, c, d} B = {d, c, a, b}

Solution: A = {a, b, c, d} and B = {d, c, a, b}. In sets, the order of elements matter.

Both sets contain exactly the same elements. Therefore, A B.

(ii) A = {4, 8, 12, 16} B = {8, 4, 16, 18}

Solution: A = {4, 8, 12, 16} and B = {8, 4, 16, 18}. On comparison, we see that: Element is in A but not in B while element is in B but not in A._

Therefore, A B._

(iii) A = {2, 4, 6, 8, 10} B = {x : x is a positive even integer and x < 10}

Solution: A = {2, 4, 6, 8, 10}. For set B: x is a positive even integer and x < 10. Therefore, B = { , , , }._

Comparing: Element is in A but not in B. Therefore, A B.

(iv) A = {x : x is a multiple of 10} B = {10, 15, 20, 25, 30, ...}

Solution: A = {x : x is a multiple of 10} = {, , , , , ...} while B = {10, 15, 20, 25, 30, ...}

Comparing we get: A contains only multiples of while B contains numbers like and which are not multiples of 10. Therefore, A B._

4. State the reasons for the following:

(i) {1, 2, 3, ..., 10} ≠ {x : x ∈ N and 1 < x < 10}

Solution:Left side: {1, 2, 3, ..., 10}. This set contains: 1, 2, 3, , , , , , , 10. Right side: {x : x ∈ N and 1 < x < 10}. This set contains: , , , , , , ,

Reason: The left set includes and , while the right set excludes both and due to the strict inequality (< instead of ≤).

(ii) {2, 4, 6, 8, 10} ≠ {x : x = 2n+1 and x ∈ N}

Solution: Left side: {2, 4, 6, 8, 10}. These are natural numbers.
Right side: {x : x = 2n + 1 and x ∈ N}. We get: For n = 0: x = 2(0) + 1 = ; For n = 1: x = 2(1) + 1 = ; For n = 2: x = 2(2) + 1 = ; For n = 3: x = 2(3) + 1 = . We see that the formula x = 2n + 1 generates natural numbers.

Reason: The left set contains numbers while the right set contains numbers.

(iii) {5, 15, 30, 45} ≠ {x : x is a multiple of 15}

Solution: Left side: {5, 15, 30, 45} while Right side: {x : x is a multiple of 15} meaning that the set contains: { , , , , , ... }. This also includes: (since 15 × = )

Reason: is in the left set but is not a multiple of 15 while is a multiple of 15 but not in the left set. Also the right set is while the left set is .

(iv) {2, 3, 5, 7, 9} ≠ {x : x is a prime number}

Solution: Left side: {2, 3, 5, 7, 9} while Right side: {x : x is a prime number}: {, , , , , , , , , ...}

Reason: is in the left set but is not a prime number. Prime numbers like , , etc. are missing from the left set. The right set is while the left set is .

5. List all the subsets of the following sets.

(i) B = {p, q}

Solution: For a set with n elements, there are subsets. B = {p, q} has elements, so it has 22 = subsets. Subsets of B: (1) Empty set: (2) Single element subsets: { } and { } (3) The set itself: { , } (Enter the letters in alphabetical order)

Complete list: φ, {p}, {q}, {p, q}

(ii) C = {x, y, z}

Solution: C = {x, y, z} has elements, so it has subsets. Subsets of C: (1) Empty set: (2) Single element subsets: { }, { } and { } (3) Two element subsets: { , } , { , } and { , } (4) The set itself: { , , } (Enter the letters in alphabetical order)

Complete list: φ, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z}

(iii) D = {a, b, c, d}

Solution: D = {a, b, c, d} has elements, so it has subsets. Subsets of D: (1) Empty set: (2) Single element subsets: {}, {}, {}, {} (3) Two element subsets: {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d} (4) Three element subsets: { , , }, { , , }, { , , }, { , , } (5) The set itself: { , , , } (Enter the letters in alphabetical order)

Complete list: φ, {a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}, {a, b, c, d}

(iv) E = {1, 4, 9, 16}

Solution: E = {1, 4, 9, 16} has elements, so it has subsets. Subsets of E: (1) Empty set: (2) Single element subsets: {}, {}, {}, {} (3) Two element subsets: {, }, {, }, {, }, {, }, {, }, {, } (4) Three element subsets: { , , }, { , , }, { , , }, { , , } (5) The set itself: { , , , } (Enter the numbers in numerical order)

Complete list: φ, {1}, {4}, {9}, {16}, {1, 4}, {1, 9}, {1, 16}, {4, 9}, {4, 16}, {9, 16}, {1, 4, 9}, {1, 4, 16}, {1, 9, 16}, {4, 9, 16}, {1, 4, 9, 16}

(v) F = {10, 100, 1000}

Solution: F = {10, 100, 1000} has elements, so it has subsets. Subsets of F: (1) Empty set: (2) Single element subsets: {}, {} and {} (3) Two element subsets: { , }, { , }, { , } (4) The set itself: { , , } (Enter the numbers in numerical order)

Complete list: φ, {10}, {100}, {1000}, {10, 100}, {10, 1000}, {100, 1000}, {10, 100, 1000}

Chapter 2: Sets

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Exercise 2.3

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Chapter 2: Sets

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Exercise 2.3