Find the values of a and b, when:

(i) (a + 3, b –2) = (5, 1)

(ii) (a + b, 2b – 3) = (4, –5)

If A = {9, 1} and B = {1, 2, 3}, show that A × B ≠ B × A.

If P = {a, b} and Q = {x, y, z}, show that P × Q ≠ Q × P.

If A = {2, 3, 5} and B = {5, 7}, find:

(i) A × B

(ii) B × A

(iii) A × A

(iv) B × B

If A = {x ϵ N : x ≤ 3} and {x ϵ W : x < 2}, find (A × B) and (B × A). Is (A × B) = (B × A)?

If A = {1, 3, 5) B = {3, 4} and C = {2, 3}, verify that:

(i) A × (B ∪ C) = (A × B) ∪ (A × C)

(ii) A × (B ∩ C) = (A × B) ∩ (A × C)

Let A = {x ϵ W : x < 2}, B = {x ϵ N : 1 < x ≤ 4} and C = {3, 5}. Verify that:

(i) A × (B ∪ C) = (A × B) ∪ (A × C)

(ii) A × (B ∩ C) = (A × B) ∩ (A × C)

If A × B = {(–2, 3), (–2, 4), (0, 4), (3, 3), (3, 4), find A and B.

Let A = {2, 3} and B = {4, 5}. Find (A × B). How many subsets will (A × B) have?

Let A × B = {(a, b): b = 3a – 2}. if (x, –5) and (2, y) belong to A × B, find the values of x and y.

Let A and B be two sets such that n(A) = 3 and n(B) = 2.

If a ≠ b ≠ c and (a, 0), (b, 1), (c, 0) is in A × B, find A and B.

Let A = {–2, 2} and B = (0, 3, 5). Find:

(i) A × B

(ii) B × A

(iii) A × A

(iv) B × B

If A = {5, 7), find (i) A × A × A.

Let A = {–3, –1}, B = {1, 3) and C = {3, 5). Find:

(i) A × B

(ii) (A × B) × C

(iii) B × C

(iv) A × (B × C)

For any sets A, B and C prove that:

A × (B ∪ C) = (A × B) ∪ (A × C)

For any sets A, B and C prove that:

A × (B ∩ C) = (A × B) ∩ (A × C)

For any sets A, B and C prove that:

A × (B – C) = (A × B) – (A × C)

For any sets A and B, prove that

(A × B) ∩ (B × A) = (A ∩ B) × (B ∩ A).

If A and B are nonempty sets, prove that

A × B = B × A ⇔ A = B

(i) If A ⊆ B, prove that A × C ⊆ B × C for any set C.

(ii) If A ⊆ B and C ⊆ D then prove that A × C ⊆ B × D.

If A × B ⊆ C × D and A × B ≠ ϕ, prove that A ⊆ C and B ⊆ D.

If A and B be two sets such that n(A) = 3, n(B) = 4 and n(A ∩ B) = 2 then find.

(i) n(A × B)

(ii) n(B × A)

(iii) n(A × B) ∩ (B × A)

For any two sets A and B, show that A × B and B × A have an element in common if and only if A and B have an element in common.

Let A = {1, 2} and B = {2, 3}. Then, write down all possible subsets of A × B.

Let A = {a, b, c, d}, B = {c, d, e} and C = {d, e, f, g}. Then verify each of the following identities:

(i) A × (B ∩ C) = (A × B) ∩ (A × C)

(ii) A × (B – C) = (A × B) – (A × C)

(iii) (A × B) ∩ (B × A) = (A ∩ B) × (A ∩ B)

Let A and B be two nonempty sets.

(i) What do you mean by a relation from A to B?

(ii) What do you mean by the domain and range of a relation?

Find the domain and range of each of the relations given below:

(i) R = {(–1, 1), (1, 1), (–2, 4), (2, 4), (2, 4), (3, 9)}

Let A = {1, 3, 5, 7} and B = {2, 4, 6, 8}.

Let R = {(x, y), : x ϵ A, y ϵ B and x > y}.

(i) Write R in roster form.

(ii) Find dom (R) and range (R).

(iii) Depict R by an arrow diagram.

Let A = {2, 4, 5, 7} and b = {1, 2, 3, 4, 5, 6, 7, 8}.

Let R = {(x, y) x ϵ A, y ϵ B and x divides y}.

(i) Write R in roster form.

(ii) Find dom (R) and range (R).

Let A = {2, 3, 4, 5} and B = {3, 6, 7, 10}.

Let R = {(x, y): x ϵ A, y ϵ B and x is relatively prime to y}.

(i) Write R in roster form.

(ii) Find dom (R) and range (R).

Let A = {1, 2, 3, 5} AND B = {4, 6, 9}.

Let R = {(x, y): x ϵ A, y ϵ B and (x – y) is odd}.

Write R in roster form.

Let A = {(x, y): x + 3y = 12, x ϵ N and y ϵ N}.

(i) Write R in roster form.

(ii) Find dom (R) and range (R).

Let A = {1, 2, 3, 4, 5, 6}.

Define a relation R from A to A by R = {(x, y): y = x + 1}.

(i) Write R in roster form.

(ii) Find dom (R) and range (R).

(iii) What is its co-domain?

(iv) Depict R by using arrow diagram.

Let R = {(x, x + 5): x ϵ {9, 1, 2, 3, 4, 5}}.

(i) Write R in roster form.

(ii) Find dom (R) and range (R).

Let A = {1, 2, 3, 4, 6} and R = {(a, b) : a, b ϵ A, and a divides b}.

(i) Write R in roster form.

(ii) Find dom (R) and range (R).

Define a relation R from Z to Z, given by

R = {(a, b): a, b ϵ Z and (a – b) is an integer.

Find dom (R) and range (R).

Let R = {(x, y): x, y ϵ Z and x² + y² ≤ 4}.

(i) Write R in roster form.

(ii) Find dom (R) and range (R).

Let A = {2, 3} and B= {3, 5}

(i) Find (A × B) and n(A × B).

(ii) How many relations can be defined from A to B?

Let A = {3, 4} and B = {7, 9}. Let R = {(a, b): a ϵ A, b ϵ B and (a – b) is odd}.

Show that R is an empty relation from A to B.

What do you mean by a binary relation on a set A? Define the domain and range of relation on A.

Let A = {2, 3, 5} and R = {(2, 3), (2, 5), (3, 3), (3, 5)}. Show that R is a binary relation on A. Find its domain and range.

Let A = {0, 1, 2, 3, 4, 5, 6, 7, 8} and let R = {(a, b) : a, b ϵ A and 2a + 3b = 12}.

Express R as a set of ordered pairs. Show that R is a binary relation on A. Find its domain and range.

If R is a binary relation on a set A define R^–1 on A.

Let R = {(a, b) : a, b ϵ W and 3a + 2b = 15} and 3a + 2b = 15}, where W is the set of whole numbers.

Express R and R^–1 as sets of ordered pairs.

Show that (i) dom (R) = range (R^–1) (ii) range (R) = dom (R^–1)

What is an equivalence relation?

Show that the relation of ‘similarity’ on the set S of all triangles in a plane is an equivalence relation.

Let R = {(a, b) : a, b ϵ Z and (a – b) is even}.

Then, show that R is an equivalence relation on Z.

Let A = {1, 2, 3} and R = {(a, b) : a, b ϵ A and |a² – b²| ≤ 5.

Write R as a set of ordered pairs.

Mention whether R is (i) reflexive (ii) symmetric (iii) transitive. Give reason in each case.

Let R = {(a, b) : a, b ϵ Z and b = 2a – 4}. If (a, –2} ϵ R and (4, b²) ϵ R. Then, write the values of a and b.

Let R be a relation on Z, defined by (x, y) ϵ R ↔ x² + y² = 9. Then, write R as a set of ordered pairs. What is its domain?

Let A be the set of first five natural numbers and let R be a relation on A, defined by (x, y) ϵ R ↔ x ≤ y.

Express R and R^–1 as sets of ordered pairs.

Find: dom (R^–1) and range (R).

Let R = (x, y) : x, y ϵ Z and x² + y² = 25}.

Express R and R^–1 as sets of ordered pairs. Show that R = R^–1.

Let A = {a, b}. List all relation on A and find their number.

Let R = {(a, b) : a, b, ϵ N and a < b}.

Show that R is a binary relation on N, which is neither reflexive nor symmetric. Show that R is transitive.

Let A and B be two sets such that n(A) = 5, n(B) = 3 and n(A ∩ B) = 2.

(i) n(A ∪ B)

(ii) n(A × B)

(iii) n(A × B) ∩ (B × A)

Find a and b when (a – 2b, 13) = (7, 2a – 3b)

If A = {1, 2}, find A × A × A.

If A = {2, 3, 4} and B = {4, 5}, draw an arrow diagram represent (A × B}.

If A = {3, 4}, B = {4, 5} and C = {5, 6}, find A × (B × C).

If A ⊆ B, prove that A × C = B × C

Prove that A × B = B × A ⇒ A = B.

If A = {5} and B = {5, 6}, write down all possible subsets of A × B.

Let R = {(x, x²) : x is a prime number less than 10}.

(i) Write R in roster form.

(ii) Find dom (R) and range (R).

Let A = (1, 2, 3} and B = {4}

How many relations can be defined from A to B.

Let A = {3, 4, 5, 6} and R = {(a, b) : a, b ϵ A and a <b

(i) Write R in roster form.

(ii) Find: dom (R) and range (R)

(iii) Write R^–1 in roster form

Let R = {(a, b) : a, b, ϵ N and a < b}.

Show that R is a binary relation on N, which is neither reflexive nor symmetric. Show that R is transitive.