⛲ Example 0. Consider these three sets
A=the set of all even numbers, B={2,4,6}, C={2,3,4,6}
Here B⊂A since every element of B is also an even number, so is an element of A.
More formally,we could say B⊂A since if x∊B,then x∊A.
It is also true that B⊂C.
C is not a subset of A, since C contains an element, 3, that is not contained in A.
Let D={} and E={1,2,3,4}. Is D⊂E? To show D⊄E, you must find at least one element of set D that is not an element of set E. Because this cannot be done, D⊂E must be true. Using the same reasoning. we can show that the empty set is a subset of every set, including itself.
⛲ Question 1. Let F, G and H be three sets. If F∊G and G⊂H, is it true that F⊂H?. If not, give an example.
✍ Solution:
No. Let F={1}, G={{1},2} and H={{1},2,3}. Here F∊G as F={1} and G⊂H. But F⊄H as 1∊F and 1∉H.
Note that an element of a set can never be a subset of itself.⛲ Ex1. In each of the following, determine whether the statement is true or false. If it is true, then prove it. If it is false, then give an example.
(i) If x∊J and J∊K, then x∊K.
(ii) If J⊂K and K∊L, then J∊L.
✍ Solution:
(i) False,
Let J={2}, K={{2},3}
Clearly, 2∊J and J∊K, but 2∉K.
So, x∊J and J∊K need not imply that x∊K.
(ii) False, Let J={2}, K={2,3} and L={{2,3},4}
Clearly, J⊂K and K∊L, but J∉L.
Thus, J⊂K Band K∊L need not imply that J∊L.
⛲ Q2. Let M={a,b,{c,d},e}. Which of the following statements is/are true?
(i) {c,d}∊M (ii) {{c,d}⊂M
✍ Solution:
Given, M={a,b,{c,d},e}
(i) Since a,b,{c,d} and e are elements of M.
∴ {c,d}∊M
Hence, it is a true statement.
(ii) As {c,d}∊M and {{c,d}} represents a set,which is a subset of M.
∴ {{c,d}}⊂M
Hence, it is a true statement.
⛲ Ex2. If N={3,{4,5},6}, then find which of the following statements are true.
(i) {4,5}⊂N (ii) {4,5}∊N (iii) Ø⊂N (iv) {3,6}⊂N
✍ Solution:
Given, N={3,{4,5},6}
Here, 3, {4,5}, 6 all are elements of N.
(i) {4,5}⊂N, which is not true. Since, element of any set is not a subset of any set and here {4,5} is an element of N.
(ii) {4,5}∊N, is a true statement.
(iii) It is always true that Ø⊂N
(iv) {3,6} makes a set,so it is a subset of N i.e., {3,6}⊂N
⛲ Q3. is each either Element or Proper Subset? But not as Both!
Let P={1,2,{3,4},5}. Which of the following statements are incorrect and why?
(i) {3,4}⊂P
(ii) {3,4}∊P
(iii) {{3,4}}⊂P
(iv) 1∊P
(v) 1⊂P
(vi) {1,2,5}⊂P
(vii) {1,2,5}∊P
(viii) {1,2,3}⊂P
(ix) Ø∊P
(x) Ø⊂P
(xi) {Ø}⊂P
✍ Solution:
P={1,2,{3,4},5}
(i) The statement {3,4}⊂P is incorrect because 3∊{3,4}; however, 3∉P.
(ii) The statement {3,4}∊P is correct because {3,4} is an element of P.
(iii) The statement {{3,4}}⊂P is correct because {3,4}∊{{3,4}} and {3,4}∊P.
(iv) The statement 1∊A is correct because 1 is an element of P.
(v) The statement 1∊P is incorrect because an element of a set can never be a subset of itself.
(vi) The statement {1,2,5}⊂P is correct because each element of {1,2,5} is also an element of P.
(vii) The statement {1,2,5}∊P is incorrect because {1,2,5} is not an element of P.
(viii) The statement {1,2,3}⊂P is incorrect because 3∊{1,2,3}; however, 3∉P.
(ix) The statement Ø∊P is incorrect because Ø is not an element of P.
(x) The statement Ø⊂P is correct because Ø is a subset of every set.
(xi) The statement {Ø}⊂P is incorrect because Ø∊{Ø}; however, Ø∊P.
⛲ Q4. Let Q={1,2,{3,4},5}. Which of the following statements are incorrect and why?
(i) {3,4}⊂Q (ii) {3,4}∊Q (iii) {{3,4}}⊂Q (iv) 1∊Q (v) 1⊂Q (vi) {1,2,5}⊂Q (vii) {1,2,5}∊Q (viii) Ø⊂Q (ix) Ø∊Q (x) Ø⊂Q
✍ Solution:
We have, Q={1,2,{3,4},5}
(i) Since, {3,4} is a member of set Q.
∴ {3,4}∊Q
Hence, {3,4}}⊂Q is incorrect.
(ii) Since, {3,4} is a member of set Q. Hence, {3,4}∊Q is correct.
(iii) Since, {3,4} is a member of set Q. So,{{3,4}} is a subset of Q.
Hence, {{3,4}}⊂Q is correct.
(iv) Since, 1 is a member of Q. Hence, 1∊Q is correct.
(v) Since, 1 is a member of set Q. Hence, 1⊂Q is incorrect.
(vi) Since, 1, 2, 5 are members of set Q. So,{1,2,5} is a subset of set Q.
Hence, {1,2,5}⊂Q is correct.
(vii) Since, 1, 2 and 5 are members of set Q. So,{1,2,5} is a subset of Q.
Hence, {1,2,5}∊Q is incorrect.
(viii) Since, Ø is subset of every set. Hence, Ø⊂Q is correct.
(ix) Since, Ø is not a member of set Q. Hence, Ø∊Q is incorrect.
(x) Since, Ø is not a member of set Q. Hence,⊂Q is incorrect.
⛲ Ex3: is each either Element or Proper Subset? But not as Both!
Determine whether the following are true or false.
a) 3∊{3,4,5}
b) {3}∊{3,4,5}
c) {3}∊{{3},{4},{5}}
d) {3}⊂{3,4,5}
e) 3⊂{3,4,5}
f) {}⊂{3,4,5}
✍ Solution:
a) 3∊{3,4,5} is a true statement because 3 is an element of the set {3,4,5}.
b) {3}∊{3,4,5} is a false statement because {3} is a set, and the set {3} is not an element of the set {3,4,5}.
c) {3}∊{{3},{4},{5}} is a true statement because {3} is an element in the set. The elements of the set {{3},{4},{5}} are themselves sets.
d) {{3}⊂{3,4,5} is a true statement because every element of the first set is an element of the second set.
e) 3⊂{3,4,5} is a false statement because the 3 is not in braces, so it is not a set and thus cannot be a proper subset. The 3 is an element of the set as indicated in part (a).
f) {}⊂{3,4,5} is a true statement because the empty set is a proper subset of every set.
⛲ Ex4. Examine whether the following statements are true or false:
(I) {a,b}⊄{b,c,a}
(ii) {a,e}⊂{x:x is a vowel in the English alphabet}
(iii) {1,2,3}⊂{1,3,5}
(iv) {a}⊂{a,b,c}
(v) {a}∊(a,b,c)
(vi) {x:x is an even natural number less than 6}⊂{x:x is a natural number which divides 36}
✍ Solution:
(i) False. Each element of {a,b} is also an element of {b,c,a}.
(ii) True. a,e are two vowels of the English alphabet.
(iii) False. 2∊{1,2,3}; however, 2∉{1,3,5}
(iv) True. Each element of {a} is also an element of {a,b,c}.
(v) False. The elements of {a,b,c} are a,b,c. Therefore, {a}⊂{a,b,c}
(vi) True. {x:x is an even natural number less than 6}={2,4}
{x:x is a natural number which divides 36}={1,2,3,4,6,9,12,18,36}