problem archive
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Decide, among the following sets, which sets are subsets of one and another: A = \{ x : x \in \mathbb{R} \text{ and } x \text{ satisfy } x^2 - 8x + 12... -
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 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 \subseteq B (ii) A - B = \phi (iii) A \cup B = B (iv) A \cap B = A -
Show that if A \subseteq B, then C - B \subseteq 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 ∩ X = B ∩ X = φ and A ∪ X = B ∪ X for some set X, show that A = B. (Hints: A = A ∩ (A ∪ X), B = B ∩ (B ∪ X) and use Distribu... -
Find sets A, B and C such that A ∩ B, B ∩ C and A ∩ C are non-empty sets and A ∩ B ∩ C = φ -
Suppose A is a set of 2 colours and B is a set of 3 objects, i.e., A = {red, blue} and B = {b, c, s}, where b, c and s represent a particular bag, coa... -
If the three states, Delhi, Madhya Pradesh and Karnataka were making codes for the licence plates of vehicles, with the restriction that the code begi... -
As a final illustration, consider the two sets A = {a1, a2} and B = {b1, b2, b3, b4} (Fig 2.3). A x B = {(a1, b1), (a1, b2), (a1, b3), (a1, b4), (a2, ... -
If $(x + 1, y - 2) = (3, 1)$, find the values of $x$ and $y$. -
If $P = \{a, b, c\}$ and $Q = \{r\}$, form the sets $P \times Q$ and $Q \times P$. Are these two products equal? -
Let $A = \{1, 2, 3\}$, $B = \{3, 4\}$ and $C = \{4, 5, 6\}$. Find (i) $A \times (B \cap C)$ (ii) $(A \times B) \cap (A \times C)$ (iii) $A \times (B \... -
If \(\frac{x}{3} + 1, y - \frac{2}{3} = \left( \frac{5}{3}, 1 \right)\), find the values of \(x\) and \(y\). -
If the set \(A\) has 3 elements and the set \(B = \{3, 4, 5\}\), then find the number of elements in \(A \times B\). -
If \(G = \{7, 8\}\) and \(H = \{5, 4, 2\}\), find \(G \times H\) and \(H \times G\). -
State whether each of the following statements are true or false. If the statement is false, rewrite the given statement correctly. (i) If \(P = \{m, ...