Element ∊ or Proper Subset ⊂ — True or False Statements

⛲ Example 0. Consider these three sets
A=the set of all even numbers, B={2,4,6}, C={2,3,4,6}
Here BA since every element of B is also an even number, so is an element of A.
More formally,we could say BA since if xB,then xA.
It is also true that BC.
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 DE? To show DE, you must find at least one element of set D that is not an element of set E. Because this cannot be done, DE 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 FG and GH, is it true that FH?. If not, give an example.
✍ Solution:
No. Let F={1}, G={{1},2} and H={{1},2,3}. Here FG as F={1} and GH. But FH 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 xJ and JK, then xK.
(ii) If JK and KL, then JL.
✍ Solution:
(i) False,
Let J={2}, K={{2},3}
Clearly, 2∊J and JK, but 2∉K.
So, xJ and JK need not imply that xK.
(ii) False, Let J={2}, K={2,3} and L={{2,3},4}
Clearly, JK and KL, but JL.
Thus, JK Band KL need not imply that JL.

⛲ 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}

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