**a Subset Phenomenon**

**Taxonomy in Biology to Identify the Ladder of Life**

In biology, the science of classifying all living things is called

__taxonomy__. More than 2000 years ago, Aristotle formalized animal classification with his “ladder of life”: higher animals, lower animals, higher plants. lower piants. To- day, living organisms are classified into six kingdoms (or sets) called animalia, plantae, archaea, eubacteria, fungi, and protista. Even more general groupings of living things are made according to shared characteristics. The groupings. from most general to most specific, are kingdom, phylum, ciass, order, family, genus. and species. For example, a zebra.

__Equus burchelli__, is a member of the genus

__Equus__, as is the horse,

__Equus caballus__. Both the zebra and the horse are mem- bers of the universal set called the kingdom of animals and the same family, Equidae; they are members of different species (

__E. burchelli__and E. caballus), however.

Why)

__This is Important__

Scientists use sets to classify and categorize animals, plants, and all forms of life.

Definition: Subset

SetAis a subset of setB, symbolized byA⊆B, if and only if all the elements of setAare also elements of setB.

The symbol *A*⊆*B* indicates that “set *A* is a subset of set *B*.” The symbol ⊈ is used to indicate “is not a subset.” Thus, *A*⊈*B* indicates that set *A* is not a subset of set *B*.

To show that set

Ais not a subset of setB, we must find at least one element of setAthat is not an elentent of setB.

⛲ Example 1: A **Subset**?

Determine whether set *N* is a subset of set *O*.

a) *N*={rain, snow, sleet} *O*={rain, snow, sleet, hail}

b) *N*={q,r,s,t} *O*={q,r}

c) *N*={*x*|*x* is a yellow fruit} *O*={*x*|*x* is a red fruit}

d) *N*={vanilla, chocolate, rocky road} *O*={chocolate, vanilla, rocky road}

✍ Solution:

a) All the elements of set *N* are contained in set *O*, so *N*⊆*O*.

b) The elements s and t are in set *N* but not in set *O*. so *N*⊈*O* (*N* is not a subset of *O*). In this example. however. all the elements of set *O* are contained in set *N*; therefore. *O*⊆*N*.

c) There are fruits, such as bananas, that are in set *N* that are not in set *O*, so *N*⊈*O*.

d) All the elements of set *N* are contained in set *O*, so *N*⊆*O*. Note also that *O*⊆*N*. In fact, set *N* =set *O*.

Definition: Proper Subset

SetAis a proper subset of setB. symbolized byA⊂B, if and only if all the elements of setAare elements of setBand setA≠ setB(that is, setBmust contain at least one element not in setA).

Consider the sets *G*={red. blue, yellow} and *H*={red. orange. yellow, green, blue. violet}. Set *G* is a subset of set *H*, *G*⊆*H*, because every element of set *G* is also an element of set *H*. Set *G* is also a proper subset of set *H*. *G*⊂*H*, because set *G* and set *H* are not equal. Now consider *J*={ear. bus, train} and *K*={train, car, bus}. Set *J* is a subset of set *K*, C⊆D, because every element of set *J* is also an element of set *K*. Set *J*, however, is not a proper subset of set *K*, *J*⊄*K*. because set *J* and set *K* are equal sets.

**Proper Subset**

If *L*⊆*M* and *L*≠*M*, then *L* is called a proper subset of *M*, written as *L*⊂*M* and *M* is called __proper superset__ of *L*.

e.g., Let *L*={*x*:*x* is an even natural number}

and *M*={*x*:*x* is a natural number}.

Then, *L*={2,4,6,8,…}

and *M*={1,2,3,4,5,…}

∴

*L*⊂

*M*

⛲ Example 2: A **Proper Subset**?

Determine whether set *P* is a proper subset of set *Q*.

a) *P*={jazz, pop, hip hop} *Q*={classical, jazz, pop, rap, hip hop}

b) *P*={a,b,c,d} *Q*={a,c,b,d}

✍ Solution:

a) All the elements of set *P* are contained in set *Q*. and sets *P*and *Q* are not equal; thus, *P*⊂*Q*.

b) Set *P*=set *Q*, so *P*⊄*Q*. (However, *P*⊆*Q*.)

Every set is a subset of itself, but no set is a proper subset of itself. For all sets *P*, *P*⊆*P*, but *P*⊄*P*. For example, if *P*={1,2,3} then *P*⊆*P* because every element of set *P* is contained in set *P* , but *P*⊄*P* because set *P* =set *P*.

⛲ Ex3. Consider the following sets Ø, *R*={1,2} and *S*={1,4,8}.

Insert the following symbols ⊂ or ⊄ between each of the following pair of sets.

(i) Ø … *S* (ii) *R* … *S*

✍ Solution:

(i) Since, null set is proper subset of every set. ∴

*S*

(ii) Given,

*R*={1,2} and

*S*={1,4,8}. Since, element 2∉

*S*. ∴

*R*⊄

*S*

⛲ Ex4. Consider the sets

*T*={1,3},

*U*={1,5,9},

*V*={1,3,5,7,9}.

Insert the symbol ⊂ or ⊄ between each of the following pair of sets:

(i) Ø … *U* (ii) *T* … *U* (iii) *T* … *V* (iv) *U* … *V*

✍ Solution:

(i) Ø⊂*U* as Ø is a subset of every set.

(ii) A⊄*U* as 3∈*T* and 3∉*U*

(iii) *T*⊂*V* as 1,3∈*T* also belongs to *V*

(iv) *U*⊂*V* as each element of *U* is also an element of *V*.

⛲ Ex5. Decide,among the following sets,which sets are subsets of one and another: *W*={*x*:*x*∈ℝ and *x* satisfy *x*^{2}-8*x*+12=0}, *X*={2,4,6}, *Y*={2,4,6,8,…}, *Z*={6}.

✍ Solution:

*W*={*x*:*x*∈ℝ and *x* satisfy *x*^{2}-8*x*+12=0}

2 and 6 are the only solutions of *x*^{2}-8*x*+12=0

∴*W*={2,6}

*X*={2,4,6}, *Y*={2,4,6,8,…}, *Z*={6}

∴

*Z*⊂

*W*⊂

*X*⊂

*Y*.

Hence,

*W*⊂

*X*,

*W*⊂

*Y*,

*X*⊂

*Y*,

*Z*⊂

*W*,

*Z*⊂

*X*,

*Z*⊂

*Y*.

⛲ Ex6. Make correct statements by filling in the symbols ⊂ or ⊄ in the blank spaces:

(i) {2,3,4} … {1,2,3,4,5}

(ii) {a,b,c} … {b,c,d}

(iii) {*x*:*x* is a student of Class XI of your school} … {*x*:*x* student of your school}

(iv) {*x*:*x* is a circle in the plane} … {*x*:*x* is a circle in the same plane with radius 1 unit}

(v) {*x*:*x* is a triangle in a plane} … {*x*:*x* is a rectangle in the plane}

(vi) {*x*:*x* is an equilateral triangle in a plane} … {*x*:*x* is a triangle in the same plane}

(vii) {*x*:*x* is an even natural number} … {*x*:*x* is an integer}

✍

(i) {2,3,4}⊂{1,2,3,4,5}

(ii) {a,b,c}⊄{b,c,d}

(iii) {*x*:*x* is a student of class XI of your school}⊂{*x*:*x* is student of your school}

(iv) {*x*:*x* is a circle in the plane}⊄{*x*:*x* is a circle in the same plane with radius 1 unit}

(v) {*x*:*x* is a triangle in a plane}⊄{*x*:*x* is a rectangle in the plane}

(vi) {*x*:*x* is an equilateral triangle in a plane}⊂{*x*:*x* in a triangle in the same plane} (vii) {*x*:*x* is an even natural number}⊂{*x*:*x* is an integer}

🌈 Element ∊ or Proper Subset ⊂ — True or False Statements