Question

3. Answer the following question about cardinality: (a) A = {a}, what is |A|? (b) B = {a, a}, what is |B|? (c) C = {{a}}, what is |C|? (d) D = {a, {a}}, what is |D|? (e) E = {∅, {∅}, {∅, {∅}}}, what is |E|? (f) X = {{a, {a, b}, {c}}, {b, {a, c}, {d, e, f}}}, what is |X|?

Answer 1

Cardinality of a set is equal to the number of elements in it.

a) For A = {a}, total number of elements = 1

So, |A| = 1

b) B = {a, a}

In set there is no meaning of repetition so basically it's B = {a}

So, again number of elements = 1

So, |B| = 1

c) C = {{a}}

Element is: {a}, so number of elements = 1

|C| = 1

d) D = {a, {a}}

Elements = a, {a}

Number of elements = |D| = 2

e) E = {∅, {∅}, {∅, {∅}}}

Elements = ∅, {∅}, {∅, {∅}}

Number of elements = |E| = 3

Here we need to understand that there is a difference in ∅ and {∅}. Only ∅ shows empty set whereas {∅} shows non-empty set with one element.

f) Elements: {a, {a, b},{c}} , {b, {a, c}, {d, e,f}}

Number of elements = |X| = 2.