ClassesClass 10MathematicsSetsExercise miscellaneous

Sets

Exercise miscellaneous

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Exercise miscellaneous All Exercises

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10 total

Decide, among the following sets, which sets are subsets of one and another: A = { x : x \in R and x satisfy x2 - 8x + 12 = 0 }, B = { 2, 4, 6 }, C = { 2, 4, 6, 8, . . . }, D = { 6 }.

Find sets A, B and C such that A \cap B, B \cap C and A \cap C are non-empty sets and A \cap B \cap C = φ.

In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example. (i) If x \in A and A \in B , then x \in B (ii) If A \subset B and B \in C , then A \in C (iii) If A \subset B and B \subset C , then A \subset C (iv) If A ⊄ B and B ⊄ C , then A ⊄ C (v) If x \in A and A ⊄ B , then x \in B (vi) If A \subset B and x \notin B , then x \notin A

Let A, B, and C be the sets such that A \cup B = A \cup C and A \cap B = A \cap C. Show that B = C.

Show that the following four conditions are equivalent : (i) A \subset B(ii) A - B = φ (iii) A \cup B = B (iv) A \cap B = A

Show that if A \subset B, then C - B \subset C - A.

Show that for any sets A and B, A = ( A \cap B ) \cup ( A - B ) and A \cup ( B - A ) = ( A \cup B )

Using properties of sets, show that (i) A \cup ( A \cap B ) = A (ii) A \cap ( A \cup B ) = A.

Show that A \cap B = A \cap C need not imply B = C.

Let A and B be sets. If A \cap X = B \cap X = φ and A \cup X = B \cup X for some set X, show that A = B. (Hints A = A \cap ( A \cup X ) , B = B \cap ( B \cup X ) and use Distributive law )