Give an example to show that A ⊂ B , B ⊂ C ⇒ A ⊂ C .

A set P is a subset of another set Q if all elements of the set P are elements of the set Q . In other words, P ⊂ Q if whenever a ∈ P , then a ∈ Q . Let's consider three sets: A = 1 , 2 , B = 1 , 2 , 3 , C = 1 , 2 , 3 , 4 In this example, we have A as a subset of B , because every element in set A ( 1 and 2 ) is also present in set B . Similarly, we have B as a subset of C , because every element in set B ( 1 , 2 and 3 ) is also present in set C . Now we can observe that, every element in set A ( 1 and 2 ) is also present in C . Therefore, A ⊂ C . Thus, the transitive property of subsets allows us to conclude that A is also a subset of C .

If a ∈ A and a ∈ B , does it follow that A ⊆ B ? Give reasons.

No, the fact that an element " a " belongs to both set A and set B does not necessarily imply that A is a subset of B . To establish that A is a subset of B , it must be shown that all elements of set A are also elements of set B . In other words, for every element " x " in A , " x " must also be an element of B . A could potentially contain elements that are not present in B , which would prevent A from being a subset of B . For example: A = 1 , 2 and B = 1 , 4 , 5 Element 2 is present in A but not in B . Hence, A need not be the subset of B .

If A = 2 , 3 , 4 , 6 , 7 be a given set, insert approximately symbol ⊂ , ⊄ , = , in the following blank space: 3 , 4 _____ A

A set P is a subset of another set Q if all elements of the set P are elements of the set Q . In other words, P ⊂ Q if whenever a ∈ P , then a ∈ Q . Given, A = { 2 , 3 , 4 , 6 , 7 } Clearly, 3 , 4 ⊂ A as 3 , 4 ∈ 3 , 4 and, also belongs to A . Hence, 3 , 4 ⊂ A .

If A = 2 , 3 , 4 , 6 , 7 be a given set, insert approximately symbol ⊂ , ⊄ , = , in the following blank space: 6 , 2 _____ A

A set P is a subset of another set Q if all elements of the set P are elements of the set Q . In other words, P ⊂ Q if whenever a ∈ P , then a ∈ Q . Given, A = { 2 , 3 , 4 , 6 , 7 } Clearly, 6 , 2 ⊂ A as 6 , 2 ∈ 6 , 2 and, also belongs to A . Hence, 6 , 2 ⊂ A .

If A = 2 , 3 , 4 , 6 , 7 be a given set, insert approximately symbol ⊂ , ⊄ , = , in the following blank space: 3 , 4 , 5 _____ A

A set P is a subset of another set Q if all elements of the set P are elements of the set Q . In other words, P ⊂ Q if whenever a ∈ P , then a ∈ Q . Given, A = { 2 , 3 , 4 , 6 , 7 } Clearly, 5 ∈ 3 , 4 , 5 but 5 does not belongs to A . Hence, 3 , 4 , 5 ⊄ A .

If A = 2 , 3 , 4 , 6 , 7 be a given set, insert approximately symbol ⊂ , ⊄ , = , in the following blank space: 2 , 4 , 3 , 7 , 6 _____ A

A set is a well-defined collection of objects. Two sets are said to be equal if they have exactly the same elements otherwise the set is said to be unequal. For example: C = x : x 2 - 1 = 0 = D = - 1 , 1 We have, A = 2 , 3 , 4 , 6 , 7 Clearly, Elements of set 2 , 4 , 3 , 7 , 6 are exactly same as set A , so 2 , 4 , 3 , 7 , 6 = A

Express the following set as an interval: { x : x ∈ R , - 3 ≤ x ≤ 5 }

Given: { x : x ∈ R , - 3 ≤ x ≤ 5 } An interval in which both endpoints are included, is called a closed interval. It is denoted by using square brackets i.e., [ a , b ] , where a , b are the endpoints of the set and square bracket means endpoints are included. In the given set, - 3 and 5 are included. Therefore, the given set as an interval is [ - 3 , 5 ] .

Express the following set as an interval: { x : x ∈ R , - 7 < x < q }

Given: { x : x ∈ R , - 7 < x < q } An open interval does not include its endpoints, and is indicated with parentheses . A closed interval is an interval which includes all its end points, and is denoted with square brackets . In the given set - 7 is not included and also q is not included. Therefore, the given set as an interval is - 7 , q