Question

Using the properties and definition of set theory, find are the following sets equal? $A=\left\{ x:~x\text{ }is\text{ }a\text{ }letter\text{ }in\text{ }the\text{ }word\text{ }reap \right\}$, $B=\left\{ x:~x\text{ }is\text{ }a\text{ }letter\text{ }in\text{ }the\text{ }word\text{ }paper \right\}$ and $C=\left\{ x:~x\text{ }is\text{ }a\text{ }letter\text{ }in\text{ }the\text{ }word\text{ }rope \right\}$.

Answer 1

Hint: Two sets are equal means each and every member of the first set is a member of second set also all the members of second set are a member in first set, that is, if $A\subseteq B$ and $B\subseteq A$ then $A=B$.

Complete step-by-step answer:

It is given in the equation to find that the given sets are equal or not. We have set $A=\left\{ x:~x\text{ }is\text{ }a\text{ }letter\text{ }in\text{ }the\text{ }word\text{ }reap \right\}$, set $B=\left\{ x:~x\text{ }is\text{ }a\text{ }letter\text{ }in\text{ }the\text{ }word\text{ }paper \right\}$ and set $C=\left\{ x:~x\text{ }is\text{ }a\text{ }letter\text{ }in\text{ }the\text{ }word\text{ }rope \right\}$.
Since all the sets are in set builder form, we will convert them into roster form that is set $A=\left\{ r,e,a,p \right\}$.
If any element is repeating then it is written only one time in set form. Like in the word ‘Paper’, ‘p’ is repeated but when we write it in the set form then we have only 4 elements: a,p,e,r. We get set $B=\left\{ a,p,e,r \right\}$.
Similarly, we get set $C=\left\{ r,o,p,e \right\}$. Now, we know that two sets are equal if all the elements of A are present in set B and all the elements of B is present in set A. or we can say that A is a subset of B and B is a subset of A., that is, , if $A\subseteq B$ and $B\subseteq A$ then $A=B$.
Now we will compare all the elements in set A, set B, set C.
Set $A=\left\{ p,a,e,r \right\}$, Set $B=\left\{ a,p,e,r \right\}$, clearly $A\subseteq B$ and $B\subseteq A$ all the element of A is present in B and all the element of B is present in A, thus set A = set B.
Now, we will compare any one set among set A and set B with Set C.
Set $B=\left\{ a,p,e,r \right\}$ and set $C=\left\{ r,o,p,e \right\}$. So in set C, ‘a’ element is not present also in set B, ‘o’ element is not present it means $setC\ne setB$ and also $setC\ne setA$, thus we can say that - $setA=setB\ne setC$ also, all the set A,B,C are not equal.

Note: Usually a student repeats the two elements in a set while representing it in normal set form, which is a major mistake. Like in this question when we are converting set B from set builder to roster form ‘p’ element is representing and we have considered only one ‘p’. We have to represent word paper into set B, we have five words p,a,p,e,r but when we represent it in set we will only write $\left\{ p,a,e,r \right\}$.