Advertisements
Advertisements
Question
Let A = {1, 2, 4, 5} B = {2, 3, 5, 6} C = {4, 5, 6, 7}. Verify the following identitie:
\[A \cap \left( B ∆ C \right) = \left( A \cap B \right) ∆ \left( A \cap C \right)\]
Solution
\[A \cap \left( B ∆ C \right) = \left( A \cap B \right) ∆ \left( A \cap C \right)\]
LHS
\[(B ∆ C) = (B - C) \cup (C - B)\]
\[(B - C) = {2, 3}\]
\[(C - B) = {4, 7}\]
\[(B - C) \cup (C - B) = {2, 3, 4, 7}\]
\[ \Rightarrow (B ∆ C) = {2, 3, 4, 7}\]
\[A \cap (B ∆ C) = {2, 4}\]
RHS
\[(A \cap B) = {2, 5}\]
\[(A \cap C) = {4, 5}\]
\[(A \cap B) ∆ (A \cap C) = {(A \cap B) - (A \cap C)} \cup {(A \cap C) - (A \cap B)}\]
\[(A \cap B) - (A \cap C) = {2}\]
\[(A \cap C) - (A \cap B) = {4}\]
\[{(A \cap B) - (A \cap C)} \cup {(A \cap C) - (A \cap B)} = {2, 4}\]
\[ \Rightarrow (A \cap B) ∆ (A \cap C) = {2, 4}\]
LHS = RHS
∴ \[A \cap \left( B ∆ C \right) = \left( A \cap B \right) ∆ \left( A \cap C \right)\]
APPEARS IN
RELATED QUESTIONS
Find the union of the following pairs of sets:
A = {x : x is a natural number and multiple of 3}
B = {x : x is a natural number less than 6}
Find the union of the following pairs of sets:
A = {x : x is a natural number and 1 < x ≤ 6}
B = {x : x is a natural number and 6 < x < 10}
Find the union of the following pairs of sets:
A = {1, 2, 3}, B = Φ
Let A = {a, b}, B = {a, b, c}. Is A ⊂ B? What is A ∪ B?
If A and B are two sets such that A ⊂ B, then what is A ∪ B?
If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10}; find
A ∪ B
Is it true that for any sets A and B, P (A) ∪ P (B) = P (A ∪ B)? Justify your answer.
Show that for any sets A and B, A = (A ∩ B) ∪ (A – B) and A ∪ (B – A) = (A ∪ B)
Let A = {1, 2, 4, 5} B = {2, 3, 5, 6} C = {4, 5, 6, 7}. Verify the following identitie:
\[A \cap \left( B \cup C \right) = \left( A \cap B \right) \cup \left( A \cap C \right)\]
If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10}; find
A ∪ C
Observe the given Venn diagram and write the following sets.
- A
- B
- A ∪ B
- U
- A'
- B'
- (A ∪ B)'
Observe the Venn diagram and write the given sets U, A, B, A ∪ B, A ∩ B.
Each set Xr contains 5 elements and each set Yr contains 2 elements and \[\bigcup\limits_{r=1}^{20} X_{r} = S = \bigcup\limits_{r=1}^{n} Y_{r}\] If each element of S belong to exactly 10 of the Xr’s and to exactly 4 of the Yr’s, then n is ______.
Given L = {1, 2, 3, 4}, M = {3, 4, 5, 6} and N = {1, 3, 5}. Verify that L – (M ∪ N) = (L – M) ∩ (L – N)
Determine whether the following statement is true or false. Justify your answer.
For all sets A, B, and C, if A ⊂ C and B ⊂ C, then A ∪ B ⊂ C
For all sets A and B, A – (A ∩ B) = A – B
For all sets A and B, (A ∪ B) – B = A – B
If A and B are two sets, then A ∩ (A ∪ B) equals ______.
If X and Y are two sets and X′ denotes the complement of X, then X ∩ (X ∪ Y)′ is equal to ______.
If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 2, 3, 5}, B = {2, 4, 6, 7} and C = {2, 3, 4, 8}. Then (B ∪ C)′ is ______.
If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 2, 3, 5}, B = {2, 4, 6, 7} and C = {2, 3, 4, 8}. Then (C – A)′ is ______.
Let S1 = `{x ∈ R - {1, 2}: ((x + 2)(x^2 + 3x + 5))/(-2 + 3x - x^2) ≥ 0}` and S2 = {x ∈ R : 32x – 3x+1 – 3x+2 + 27 ≤ 0}. Then, S1 ∪ S2 is equal to ______.
