Question

definition with example​

Answer 1

Answer:

1. A set is a collection of elements or numbers or objects, represented within the curly brackets { }. For example: {1,2,3,4} is a set of numbers.
2. The union of two sets is a set containing all elements that are in A or in B (possibly both). For example, {1,2}∪{2,3}={1,2,3}. Thus, we can write x∈(A∪B) if and only if (x∈A) or (x∈B). Note that A∪B=B∪A.
3. The set operation intersection takes only the elements that are in both sets. The intersection contains the elements that the two sets have in common. The intersection is where the two sets overlap. In set-builder notation, A ∩ B = {x ∈ U : x ∈ A and x ∈ B}.
4. Set Difference: The relative complement or set difference of sets A and B, denoted A – B, is the set of all elements in A that are not in B. Example: Let A = {a, b, c, d} and B = {b, d, e}. Then A – B = {a, c} and B – A = {e}.
5. Complement of a Set: The complement of a set, denoted A', is the set of all elements in the given universal set U that are not in A. Example: U' = ∅ The complement of the universe is the empty set. Example: ∅' = U The complement of an empty set is the universal set.
6. The number of distinct elements in a finite set is called its cardinal number. It is denoted as n(A) and read as 'the number of elements of the set'. For example: (i) Set A = {2, 4, 5, 9, 15} has 5 elements. Therefore, the cardinal number of set A = 5.
7. Subsets are a part of one of the mathematical concepts called Sets. A set is a collection of objects or elements, grouped in the curly braces, such as {a,b,c,d}. If a set A is a collection of even number and set B consists of {2,4,6}, then B is said to be a subset of A, denoted by B⊆A and A is the superset of B.

Types of Subsets

Subsets are classified as

Proper Subset

Improper Subsets

A proper subset is one that contains a few elements of the original set whereas an improper subset, contains every element of the original set along with the null set.

For example, if set A = {2, 4, 6}, then,

Number of subsets: {2}, {4}, {6}, {2,4}, {4,6}, {2,6}, {2,4,6} and Φ or {}.

Proper Subsets: {}, {2}, {4}, {6}, {2,4}, {4,6}, {2,6}

Improper Subset: {2,4,6}

There is no particular formula to find the subsets, instead, we have to list them all, to differentiate between proper and improper one. The set theory symbols were developed by mathematicians to describe the collections of objects.

8.A universal set is a set which contains all the elements or objects of other sets, including its own elements. It is usually denoted by the symbol 'U'. Suppose Set A consists of all even numbers such that, A = {2, 4, 6, 8, 10, …} and set B consists of all odd numbers, such that, B = {1, 3, 5, 7, 9, …}.

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Answer 2

1. ɪɴ ᴍᴀᴛʜᴇᴍᴀᴛɪᴄs ᴀ sᴇᴛ ɪs ᴀ ᴄᴏʟʟᴇᴄᴛɪᴏɴ ᴏғ ᴅɪsᴛɪɴᴄᴛ ᴇʟᴇᴍᴇɴᴛs. ᴛʜᴇ ᴇʟᴇᴍᴇɴᴛs ᴛʜᴀᴛ ᴍᴀᴋᴇ ᴜᴘ ᴀ sᴇᴛ ᴄᴀɴ ʙᴇ ᴀɴʏ ᴋɪɴᴅ ᴏғ ᴛʜɪɴɢs: ᴘᴇᴏᴘʟᴇ, ʟᴇᴛᴛᴇʀs ᴏғ ᴛʜᴇ ᴀʟᴘʜᴀʙᴇᴛ, ɴᴜᴍʙᴇʀs, ᴘᴏɪɴᴛs ɪɴ sᴘᴀᴄᴇ, ʟɪɴᴇs, ᴏᴛʜᴇʀ ɢᴇᴏᴍᴇᴛʀɪᴄᴀʟ sʜᴀᴘᴇs, ᴠᴀʀɪᴀʙʟᴇs, ᴏʀ ᴇᴠᴇɴ ᴏᴛʜᴇʀ sᴇᴛs. ᴛᴡᴏ sᴇᴛs ᴀʀᴇ ᴇǫᴜᴀʟ ɪғ ᴀɴᴅ ᴏɴʟʏ ɪғ ᴛʜᴇʏ ʜᴀᴠᴇ ᴘʀᴇᴄɪsᴇʟʏ ᴛʜᴇ sᴀᴍᴇ ᴇʟᴇᴍᴇɴᴛs.

2. ɪɴ sᴇᴛ ᴛʜᴇᴏʀʏ, ᴛʜᴇ ᴜɴɪᴏɴ ᴏғ ᴀ ᴄᴏʟʟᴇᴄᴛɪᴏɴ ᴏғ sᴇᴛs ɪs ᴛʜᴇ sᴇᴛ ᴏғ ᴀʟʟ ᴇʟᴇᴍᴇɴᴛs ɪɴ ᴛʜᴇ ᴄᴏʟʟᴇᴄᴛɪᴏɴ. ɪᴛ ɪs ᴏɴᴇ ᴏғ ᴛʜᴇ ғᴜɴᴅᴀᴍᴇɴᴛᴀʟ ᴏᴘᴇʀᴀᴛɪᴏɴs ᴛʜʀᴏᴜɢʜ ᴡʜɪᴄʜ sᴇᴛs ᴄᴀɴ ʙᴇ ᴄᴏᴍʙɪɴᴇᴅ ᴀɴᴅ ʀᴇʟᴀᴛᴇᴅ ᴛᴏ ᴇᴀᴄʜ ᴏᴛʜᴇʀ. ᴀ ɴᴜʟʟᴀʀʏ ᴜɴɪᴏɴ ʀᴇғᴇʀs ᴛᴏ ᴀ ᴜɴɪᴏɴ ᴏғ ᴢᴇʀᴏ sᴇᴛs ᴀɴᴅ ɪᴛ ɪs ʙʏ ᴅᴇғɪɴɪᴛɪᴏɴ ᴇǫᴜᴀʟ ᴛᴏ ᴛʜᴇ ᴇᴍᴘᴛʏ sᴇᴛ.

3. ɪɴ ᴍᴀᴛʜᴇᴍᴀᴛɪᴄs, ᴛʜᴇ ɪɴᴛᴇʀsᴇᴄᴛɪᴏɴ ᴏғ ᴛᴡᴏ sᴇᴛs ᴀ ᴀɴᴅ ʙ, ᴅᴇɴᴏᴛᴇᴅ ʙʏ ᴀ ∩ ʙ, ɪs ᴛʜᴇ sᴇᴛ ᴄᴏɴᴛᴀɪɴɪɴɢ ᴀʟʟ ᴇʟᴇᴍᴇɴᴛs ᴏғ ᴀ ᴛʜᴀᴛ ᴀʟsᴏ ʙᴇʟᴏɴɢ ᴛᴏ ʙ (ᴏʀ ᴇǫᴜɪᴠᴀʟᴇɴᴛʟʏ, ᴀʟʟ ᴇʟᴇᴍᴇɴᴛs ᴏғ ʙ ᴛʜᴀᴛ ᴀʟsᴏ ʙᴇʟᴏɴɢ ᴛᴏ ᴀ).

4. sᴇᴛ ᴅɪғғᴇʀᴇɴᴄᴇ: ᴛʜᴇ ʀᴇʟᴀᴛɪᴠᴇ ᴄᴏᴍᴘʟᴇᴍᴇɴᴛ ᴏʀ sᴇᴛ ᴅɪғғᴇʀᴇɴᴄᴇ ᴏғ sᴇᴛs ᴀ ᴀɴᴅ ʙ, ᴅᴇɴᴏᴛᴇᴅ ᴀ – ʙ, ɪs ᴛʜᴇ sᴇᴛ ᴏғ ᴀʟʟ ᴇʟᴇᴍᴇɴᴛs ɪɴ ᴀ ᴛʜᴀᴛ ᴀʀᴇ ɴᴏᴛ ɪɴ ʙ. ɪɴ sᴇᴛ-ʙᴜɪʟᴅᴇʀ ɴᴏᴛᴀᴛɪᴏɴ, ᴀ – ʙ = {x ∈ ᴜ : x ∈ ᴀ ᴀɴᴅ x ∉ ʙ}= ᴀ ∩ ʙ'. ... ᴛʜᴇɴ ᴛʜᴇ sᴇᴛ ᴅɪғғᴇʀᴇɴᴄᴇ ᴏғ ᴀ ᴀɴᴅ ʙ ᴡᴏᴜʟᴅ ʙᴇ ᴛʜᴇ $407 ʀᴇᴍᴀɪɴɪɴɢ ɪɴ ᴛʜᴇ ᴄʜᴇᴄᴋɪɴɢ ᴀᴄᴄᴏᴜɴᴛ.

5. ᴄᴏᴍᴘʟᴇᴍᴇɴᴛ ᴏғ ᴀ sᴇᴛ: ᴛʜᴇ ᴄᴏᴍᴘʟᴇᴍᴇɴᴛ ᴏғ ᴀ sᴇᴛ, ᴅᴇɴᴏᴛᴇᴅ ᴀ', ɪs ᴛʜᴇ sᴇᴛ ᴏғ ᴀʟʟ ᴇʟᴇᴍᴇɴᴛs ɪɴ ᴛʜᴇ ɢɪᴠᴇɴ ᴜɴɪᴠᴇʀsᴀʟ sᴇᴛ ᴜ ᴛʜᴀᴛ ᴀʀᴇ ɴᴏᴛ ɪɴ ᴀ. ɪɴ sᴇᴛ- ʙᴜɪʟᴅᴇʀ ɴᴏᴛᴀᴛɪᴏɴ, ᴀ' = {x ∈ ᴜ : x ∉ ᴀ}. ... ᴇxᴀᴍᴘʟᴇ: ᴜ' = ∅ ᴛʜᴇ ᴄᴏᴍᴘʟᴇᴍᴇɴᴛ ᴏғ ᴛʜᴇ ᴜɴɪᴠᴇʀsᴇ ɪs ᴛʜᴇ ᴇᴍᴘᴛʏ sᴇᴛ. ᴇxᴀᴍᴘʟᴇ: ∅' = ᴜ ᴛʜᴇ ᴄᴏᴍᴘʟᴇᴍᴇɴᴛ ᴏғ ᴀɴ ᴇᴍᴘᴛʏ sᴇᴛ ɪs ᴛʜᴇ ᴜɴɪᴠᴇʀsᴀʟ sᴇᴛ.