Discrete Structures 1. List the members of these sets. a) { x | x is the square

Question

Discrete Structures

1. List the members of these sets.
a) {x | x is the square of an integer and x < 100}

b) {x | x is an integer such that x2 = 2}

2. Use set builder notation to give a description of each of these sets. a) {3,2,1, 0, 1, 2, 3}

b) {m, n, o, p}

3. Determine whether each of these pairs of sets are equal. a) {1, 3, 3, 3, 5, 5, 5, 5, 5}, {5, 3, 1}
b) {{1}}, {1, {1}}

c) , {}
4. Suppose that A = {2, 4, 6}, B = {2, 6}, C = {4, 6}, and D = {4, 6, 8}. Determine which of

these sets are subsets of which other of these sets.

5. For each of the following sets, determine whether 2 is an element of that set. a) {x R | x is an integer greater than 1}
b) {x R | x is the square of an integer}
c) {2,{2}}

d) {{2},{{2}}} e) {{2},{2,{2}}} f ) {{{2}}}

6. Determine whether each of these statements is true or false. a) 0
b) {0}
c) {0}

d) {0} e) {0} {0} f ) {0} {0} g) {} {}

7. Determine whether these statements are true or false. a) {}
b) {, {}}
c) {} {}

d) {} {{}}
e) {} {, {}}
f ) {{}} {, {}} g) {{}} {{}, {}}

8. What is the cardinality of each of these sets? a) { }

b) {a, {a}}

c) {a, {a}, {a, {a}}}

9. Find the power set of each of these sets, where a and b are distinct elements. a) {a}

b) {a,b}
10. Let A = {a, b, c}, B = {x, y}, and C = {0, 1}. Find A × B × C.

11.FindA3 if A={a,1}
12. How many different elements does A × B × C have if A has m elements, B has n

elements, and C has p elements?

13. Let A = {0, 2, 4, 6, 8, 10}, B = {0, 1, 2, 3, 4, 5, 6}, and C = {4, 5, 6, 7, 8, 9, 10}. Find a) (A B)C.

b) (AB) C.

14. Draw the Venn diagrams for the combination of the sets A, B, and C A(BC)

15. Draw the Venn diagrams for the combination of the sets A, B, and C. a) A(BC)

Explanation / Answer

1. List the members of these sets.

a) {x | x is the square of an integer and x < 100}

here we have to list all the squares of an integer whose value < 100

i.e., {0,1,4,9,16,25,36,49,64,81}

b) {x | x is an integer such that x2 = 2}

there is no element when we power an integer by 2 result 2 so it is empty set.

2. Use set builder notation to give a description of each of these sets.

a) {3,2,1, 0, 1, 2, 3}

given is a range of integers from -3 to 3 so the set buider form of it is
{x| 3x3}, where the domain is the set of integers.

b) {m, n, o, p}

given is a set containg letters m,n,o,p so i choose word topman to describe the set
{x|xis a letter of the word "topman" other than a or t}

3. Determine whether each of these pairs of sets are equal.

a) {1, 3, 3, 3, 5, 5, 5, 5, 5}, {5, 3, 1}

yes beacuse set consist only unique elements so when we remove duplicate elements from first set then both are equal.

b) {{1}}, {1, {1}}

no because in the 1'st set we have only 1 element which is nested inside another set but in 2'nd set we have 2 elements so both are not equal.

c) , {}

no because represents empty set but {} represents a set containing 1 element i.e., empty set.

4. Suppose that A = {2, 4, 6}, B = {2, 6}, C = {4, 6}, and D = {4, 6, 8}. Determine which of
these sets are subsets of which other of these sets.

subset means a set which conains elements of other set so when we seeing the given sets every set contain elements of other set so

Every set is subset of itself
B is a subset of A and
C is a subset of both A and D

5. For each of the following sets, determine whether 2 is an element of that set.

a) {x R | x is an integer greater than 1}

yes 2 is an element of this set because 2 is greater than 1

b) {x R | x is the square of an integer}

no 2 is not a square of an integer beacause square of an interger never result 2

c) {2,{2}}

yes 2 is in the given set

d) {{2},{{2}}} = no there is no 2 in the list of elements only sets contain 2 but those sets are not equal to 2

e) {{2},{2,{2}}} = no there is no 2 in the list of elements only sets contain 2 but those sets are not equal to 2

f) {{{2}}} = no there is no 2 in the list of elements only sets contain 2 but those sets are not equal to 2

6. Determine whether each of these statements is true or false.

a) 0 = false

0 is an element not empty so it is false

b) {0} = false

it is an 1 element set so it is not empty so it is also false

c) {0} = false

a set with 1 element is not a subset of empty set so it is also false

d) {0} = true

true because an empty set is always a subset of any set

e) {0} {0} = false

the same sets are not belong to each other so it is false

f ) {0} {0} = false

the same set are not subsets of each other so it is also false

g) {} {} = true

the sets are same with belongs to relationship so given statement is true

7. Determine whether these statements are true or false.

a) {} = true
b) {, {}} = true
c) {} {} = false
d) {} {{}} = true
e) {} {, {}} = true
f ) {{}} {, {}} = false
g) {{}} {{}, {}} = flase

8. What is the cardinality of each of these sets?

cardinality = number of elements in a set

a) { } = 0
b) {a, {a}} = 2
c) {a, {a}, {a, {a}}} = 3

9. Find the power set of each of these sets, where a and b are distinct elements.

the power set of any set S is the set of all subsets of S, including the empty set and S itself

a) {a} = {,{a}}
b) {a,b} = {,{a},{b},{a,b}}

10. Let A = {a, b, c}, B = {x, y}, and C = {0, 1}. Find A × B × C.

Cartesian product A × B = the set of all ordered pairs (a, b) where a A and b B.

{(a,x,0),(a,x,1),(a,y,0),(a,y,1),(b,x,0),(b,x,1),(b,y,0),(b,y,1),(c,x,0),(c,x,1),(c,y,0),(c,y,1)}

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