Let U = {1, 2, 3, 4, 5, 6, 7} be the universal set and let A = {1, 3, 5} and B = {2, 4} be two of its disjoint subsets. Using Venn diagrams show that \[A\cap B=\phi \].
VerifiedHint: First of all, recollect all the terms like a universal set, subsets, null set, etc. Now, draw a Venn diagram for universal set U. Now, inside it draws the subsets A and B and shows that \[A\cap B=\phi \].
Complete step-by-step answer:
Here, we are given that the universal set U = {1, 2, 3, 4, 5, 6, 7} and other two sets A = {1, 3, 5} and B = {2, 4} as the subsets of U. Here, we have to show that \[A\cap B=\phi \] by Venn diagram. Before proceeding with this question, let us consider a few terms.
Universal set: A universal set is a set which contains all the elements or objects of other sets, including its own elements. It is usually denoted by the symbol ‘U’. For example, if we have 3 sets A = {1, 3, 6, 8}, B = {2, 3, 4, 5}, C = {5, 8, 9} then our universal set is U = {1, 2, 3, 4, 5, 6, 8, 9}
Subset: A set A is a subset of another set B if all elements of the set A are elements of the set B. In other words, the set A is contained inside the set B. The subset relationship is denoted as \[A\subset B\]. For example, if we have a set A = {1, 3, 6, 8} then some subsets of set A are {1, 6,}, {1, 3,}, {1, 3, 8,} etc.
Null set: Any Set that does not contain any element is called the empty or null or void set. The symbol used to represent an empty set is { } or \[\phi \]. For example, if we have a set A = {x: 11 < x < 12, x is a natural number} then this will be a null set because there is no natural number between numbers 11 and 12.
Therefore, A = { } or \[\phi \].
The intersection of two sets: It is denoted by \[A\cap B\]. This is the set containing all elements of A that also belong to B (or equivalently, all elements of B that also belong to A). For example if we have two sets A = {1, 2, 4} and B = {3, 7, 8, 9, 0}. Then \[A\cap B=\phi \] because no element in A also belongs to B or vice versa.
Now, let us consider our question. First of all, let us draw the Venn diagram for the universal set U = {1, 2, 3, 4, 5, 6, 7}.
Now, let us draw the subsets A = {1, 3, 5} and B = {2, 4} inside the above set.
In the above figure, we can see that the Venn diagram of two sets is not intersecting anywhere. So, we can conclude that this is not an element in \[A\cap B\]. So, \[A\cap B\] is the void or empty or null set. Hence, \[A\cap B=\phi \].
Note: In this question, students must note that the subsets are always drawn inside the universal set. Also, properly take the elements inside each set according to the given information. Students should also remember the fact that \[\phi \] is a subset of every set but there is a difference between \[A=\phi \] and \[\phi \subset A\]. In the first case, A is null set and the second case is true for every set.
