Consider the sets : `X =` set of all students in your school, `Y =` set of all students in your class.
• We note that every element of `Y` is also an element of `X;` we say that `Y` is a subset of `X.`
• The fact that `Y` is subset of `X` is expressed in symbols as `Y ⊂ X.` The symbol `⊂` stands for `"‘is a subset of’"` or `"‘is contained in’."`
`\color{purple}ul(✓✓) \color{purple} " DEFINITION ALERT" `
A set `A` is said to be a subset of a set `B` if every element of `A` is also an element of `B.`
• In other words, `A ⊂ B` if whenever `a ∈ A,` then `a ∈ B.` It is often convenient to use the symbol `"“⇒”"` which means implies. Using this symbol, we can write the definition of subset as follows:
` \ \ \ \ \ \ \ \ \ \ A subset B` if `a ∈ A ⇒ a ∈ B`
• We read the above statement as “A is a subset of B. if `a` is an element of `A` implies that `a` is also an element of `B`”.
`\color{blue} (✍️ "If A is not a subset of B, we write A ⊄ B")`.
`\color{green} ✍️ \color{green} \mathbf(KEY \ CONCEPT) `
• It follows from the above definition that every set A is a subset of itself, i.e., `A subset A`.
• Since the empty set `φ` has no elements, we agree to say that `φ` is a subset of every set.
`"We now consider some examples :"`
`(i)` The set `Q` of rational numbers is a subset of the set `R` of real numbes, and we write `Q ⊂ R.`
`(ii)` If `A` is the set of all divisors of `56` and `B` the set of all prime divisors of `56,` then `B` is a subset of `A` and we write `B ⊂ A.`
`(iii)` Let `A = {1, 3, 5}` and `B = {x : x "is an odd natural number less than" 6}.` Then `A ⊂ B` and `B ⊂ A` and hence `A = B.`
`(iv)` Let `A = { a, e, i, o, u}` and `B = { a, b, c, d}.` Then `A` is not a subset of `B,` also `B` is not a subset of `A.`
`\color{green} (✍️ "Proper subset and Superset")`
Let `A` and `B` be two sets. If `A ⊂ B` and `A ≠ B` , then `A` is called a proper subset of `B` and `B` is called superset of `A.`
For example, `A = {1, 2, 3}` is a proper subset of `B = {1, 2, 3, 4}.`
`\color{green}( ✍️ "Singleton set: ")`
If a set `A` has only one element, we call it a singleton set. Thus ,`{ a }` is a singleton set.
Consider the sets : `X =` set of all students in your school, `Y =` set of all students in your class.
• We note that every element of `Y` is also an element of `X;` we say that `Y` is a subset of `X.`
• The fact that `Y` is subset of `X` is expressed in symbols as `Y ⊂ X.` The symbol `⊂` stands for `"‘is a subset of’"` or `"‘is contained in’."`
`\color{purple}ul(✓✓) \color{purple} " DEFINITION ALERT" `
A set `A` is said to be a subset of a set `B` if every element of `A` is also an element of `B.`
• In other words, `A ⊂ B` if whenever `a ∈ A,` then `a ∈ B.` It is often convenient to use the symbol `"“⇒”"` which means implies. Using this symbol, we can write the definition of subset as follows:
` \ \ \ \ \ \ \ \ \ \ A subset B` if `a ∈ A ⇒ a ∈ B`
• We read the above statement as “A is a subset of B. if `a` is an element of `A` implies that `a` is also an element of `B`”.
`\color{blue} (✍️ "If A is not a subset of B, we write A ⊄ B")`.
`\color{green} ✍️ \color{green} \mathbf(KEY \ CONCEPT) `
• It follows from the above definition that every set A is a subset of itself, i.e., `A subset A`.
• Since the empty set `φ` has no elements, we agree to say that `φ` is a subset of every set.
`"We now consider some examples :"`
`(i)` The set `Q` of rational numbers is a subset of the set `R` of real numbes, and we write `Q ⊂ R.`
`(ii)` If `A` is the set of all divisors of `56` and `B` the set of all prime divisors of `56,` then `B` is a subset of `A` and we write `B ⊂ A.`
`(iii)` Let `A = {1, 3, 5}` and `B = {x : x "is an odd natural number less than" 6}.` Then `A ⊂ B` and `B ⊂ A` and hence `A = B.`
`(iv)` Let `A = { a, e, i, o, u}` and `B = { a, b, c, d}.` Then `A` is not a subset of `B,` also `B` is not a subset of `A.`
`\color{green} (✍️ "Proper subset and Superset")`
Let `A` and `B` be two sets. If `A ⊂ B` and `A ≠ B` , then `A` is called a proper subset of `B` and `B` is called superset of `A.`
For example, `A = {1, 2, 3}` is a proper subset of `B = {1, 2, 3, 4}.`
`\color{green}( ✍️ "Singleton set: ")`
If a set `A` has only one element, we call it a singleton set. Thus ,`{ a }` is a singleton set.