Question 24 of 47beginner🔧 ApplyShort Answer4 marks

Find the intersection of each pair of sets of question 1 above.

Correct Answer

(i) The intersection of sets X={1,3,5}X = \{1, 3, 5\} and Y={1,2,3}Y = \{1, 2, 3\} is the set of elements present in both. The common elements are 1 and 3.

Therefore, X∩Y={1,3}X \cap Y = \{1, 3\}.

(ii) The intersection of sets A={a,e,i,o,u}A = \{a, e, i, o, u\} and B={a,b,c}B = \{a, b, c\} is the set of elements common to both. The only common element is 'a'.

Therefore, A∩B={a}A \cap B = \{a\}.

(iii) Set A={x:x is a natural number and multiple of 3}={3,6,9,12,… }A = \{x : x \text{ is a natural number and multiple of 3}\} = \{3, 6, 9, 12, \dots\}. Set B={x:x is a natural number less than 6}={1,2,3,4,5}B = \{x : x \text{ is a natural number less than 6}\} = \{1, 2, 3, 4, 5\}.

The element common to both sets is 3.

Therefore, A∩B={3}A \cap B = \{3\}.

(iv) Set A={x:x is a natural number and 1<x≤6}={2,3,4,5,6}A = \{x : x \text{ is a natural number and } 1 < x \leq 6\} = \{2, 3, 4, 5, 6\}. Set B={x:x is a natural number and 6<x<10}={7,8,9}B = \{x : x \text{ is a natural number and } 6 < x < 10\} = \{7, 8, 9\}.

There are no common elements between set A and set B.

Therefore, A∩B=ϕA \cap B = \phi (an empty set).

(v) Set A={1,2,3}A = \{1, 2, 3\} and set B=ϕB = \phi (an empty set).

The intersection of any set with an empty set is always an empty set as there are no elements to be common.

Therefore, A∩B=ϕA \cap B = \phi.

Exercise: EXERCISE 1.4 | Q: 5 | (Page 17)
For More Understanding

Explanation

To find the intersection of two sets, we identify elements that are present in both sets. For parts (i) and (ii), we compare the listed elements directly. For parts (iii) and (iv), we first list the elements of the sets based on their definitions (natural numbers satisfying specific conditions) and then find the common elements. In part (iii), the only multiple of 3 less than 6 is 3. In part (iv), the sets have disjoint ranges. For part (v), the intersection with an empty set is always empty.

Solution Steps

  1. Step 1: Retrieve the pairs of sets from Question 1.

  2. Step 2: For pair (i), identify common elements in XX and YY: {1,3}\{1, 3\}.

  3. Step 3: For pair (ii), identify common elements in AA and BB: {a}\{a\}.

  4. Step 4: For pair (iii), list elements of A ({3,6,...}\{3, 6, ...\}) and B ({1,2,3,4,5}\{1, 2, 3, 4, 5\}). Find common element: {3}\{3\}.

  5. Step 5: For pair (iv), list elements of A ({2,3,4,5,6}\{2, 3, 4, 5, 6\}) and B ({7,8,9}\{7, 8, 9\}). Identify no overlap: ϕ\phi.

  6. Step 6: For pair (v), identify intersection of any set with ϕ\phi: ϕ\phi.

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