**Q.1: If (a3+1,b–23) = (53,13), what is the value of a and b?**

** **

**Sol:**

As the **ordered pairs are equal**, the **corresponding elements will also be equal.**

Therefore,

**Therefore, a = 2**

Now,

**Therefore, b = 1**

**Hence, a = 2 and b = 1**

**Q:2. If the set X has 4 elements and the set Y = {2, 3, 4, 5}, then find the number of elements in X × Y**

**Sol:**

There are **4 elements** in **set X** and the **elements** of **set X** are **2, 3, 4, and 5**.

**No. of elements in X × Y = (No. of elements in X) × (No. of elements in Y)**

= 4 × 4

= 16

**Therefore, the no. of elements in (X×Y) is 16.**

**Q.3: If A = {8, 9} and B = {4, 5, 2}, what is the value of A × B and B × A?**

** **

**Sol**:

**A = {8, 9}**

**B = {4, 5, 2}**

As we know that **Cartesian product ‘P × Q’ **of two **non-empty sets ‘P’ and ‘Q’** is defined as P × Q = {(p, q): p

Therefore,

**A × B = {(8, 4), (8, 5), (8, 2), (9, 4), (9, 5), (9, 2)}**

**B × A = {(4, 8), (4, 9), (5, 8), (5, 9), (2, 8), (2, 9)}**

**Q.4: State whether the given statements are True or False. If the statement is false, write that statement correctly.**

**(i). If X = {a, b} and Y = {b, a}, then X × Y = {(a, b), (b, a)}**

**(ii). If P and Q are non – empty sets, then P × Q is a non – empty set of ordered pairs (a, b) such that x ∈ P and b ∈ Q.**

**(iii). If M = {2, 3}, N = {4, 5}, then M × (N ∩Ø ) = Ø.**

** **

**Sol:**

**(i).** If X = {a, b} and Y = {b, a}, then X × Y = {(a, b), (b, a)}

**The given statement is False.
**

**X = {a, b}**

**Y = {b, a}**

**Therefore, X × Y = {(a, b), (a, a), (b, b), (b, a)}**

** **

**(ii).** If P and Q are non – empty sets, then P × Q is a non – empty set of ordered pairs (a, b) such that x

**The given statement is True.**

**(iii).** If M = {2, 3}, N = {4, 5}, then **M × (N ∩Ø ) = Ø.**

**The given statement is True.**

**Q.5: If M = {-2, 2}, then find M × M × M.**

** **

**Sol:**

For any non – empty set ‘M’, **M × M × M** is **defined** as:

**M × M × M = {(x, y, z): x, y, z ∈ M}**

Since, M = {-2, 2} ** [ Given]**

**Therefore, M × M × M = {(–2, –2, –2), (–2, –2, 2), (–2, 2, –2), (–2, 2, 2), (2, –2, –2), (2, –2, 2), (2, 2, –2), (2, 2, 2)}**

**Q.6: If X × Y = {(a, m), (a, n), (b, m), (b, n)}. Find X and Y.**

**Sol:**

**X × Y = {(a, m), (a, n), (b, m), (b, n)}**

As we know, that **Cartesian product P × Q **of **two non-empty sets P and Q** is **defined** as **P × Q =** {(p, q): p

Therefore, ‘X’ is the set of all the **first** **elements** and ‘Y’ is the set of all the **second** **elements**.

**Therefore, X = {a, b} and Y = {m, n}**

** **

**Q.7: Let P = {2, 3}, Q = {2, 3, 4, 5}, R = {6, 7} and S = {6, 7, 8, 9}. Verify the following:**

**(i). P×(Q∩R) = (P×Q)∩(P×R)**

**(ii). P × R is a subset of Q × S**

** **

**Answer:**

**(i). **

**Taking LHS:**

= Ø

**= Ø**

**Now Taking RHS:**

**Therefore, LHS = RHS**

**(ii). P × R is a subset of Q × S**

P × R = {(2, 6), (2, 7), (3, 6), (3, 7)}

Q × S = {(2, 6), (2, 7), (2, 8), (2, 9), (3, 6), (3, 7), (3, 8), (3, 9), (4, 6), (4, 7), (4, 8), (4, 9), (5, 6), (5, 7), (5, 8), (5, 9)}

We can see that all the elements of **set P × R **are the elements of the **set Q × S.**

**Therefore, P × R is a subset of Q × S.**

**Q.8: Let P = {2, 3} and Q = {4, 5}. Find P × Q and then find how many subsets will P × Q have? List them.**

**Sol:**

**P = {2, 3}**

**Q = {4, 5}**

**P × Q = {(2, 4), (2, 5), (3, 4), (3, 5)}**

**n (P × Q) = 4**

As we know, that If ‘A’ is a set with **n(A) = m**,

Then, n[P(A)] =

Therefore,

For the **set P × Q** =

**= 16 subsets**

The **subsets** are as following:

**Ø, {(2, 4)}, {(2, 5)}, {(3, 4)}, {(3, 5)}, {(2, 4), (2, 5)}, {(2, 4), (3, 4)}, {(2, 4), (3, 5)}, {(2, 5), (3, 4)}, {(2, 5), (3, 5)}, {(3, 4), (3, 5)}, {(2, 4), (2, 5), (3, 4)}, {(2, 4), (2, 5), (3, 5)}, {(2, 4), (3, 4), (3, 5)}, {(2, 5), (3, 4), (3, 5)}, {(2, 4), (2, 5), (3, 4), (3, 5)}**

**Q.9: Let M and N be two sets where n (M) = 3 and n (N) = 2. If (a, 1), (b, 2), (c, 1) are in M × N, find M and N, where a, b and c are different elements.**

**Sol:**

**n(M) = 3**

**n(N) = 2**

Since, (a, 1), (b, 2), (c, 1) are in M × N

[M = Set of first elements of the ordered pair elements of M × N]

[N = Set of second elements of the ordered pair elements of M × N]

**Therefore, a, b and c are the elements of M **

**And, 1 and 2 are the elements of N**

**As n(M) = 3** and **n(N) = 2**, **hence M = {a, b, c} and N = {1, 2}**

**Q.10: The Cartesian product Z × Z has 9 elements among which are found (-2, 0) and (0, 2). Find the set Z and also the remaining elements of Z × Z.**

**Sol:**

As we know that, **If n(M) = p and n(N) = q, then n(M × N) = pq.**

Now,

n (Z × Z) = n(Z) × n(Z)

But, it is given that, n(Z × Z) = 9

Therefore, n(Z) × n(Z) = 9

**n(Z) = 3**

**The pairs (-2, 0) and (0, 2) are two of the nine elements of Z × Z**

As we know, Z × Z = {(x, x): x

**Therefore, –2, 0, and 2 are elements of Z**

Since, n(Z) = 3, we can see that Z = {–2, 0, 2}

**Therefore, the remaining elements of the set Z × Z are (–2, –2), (–2, 2), (0, –2), (0, 0), (2, –2), (2, 0), and (2, 2).**