Question:

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,n} and Q={n,m} then P×Q={(m,n),(n,m)} (ii) If A and B are non-empty sets then A×B is a non-empty set of ordered pairs (x,y) such that x∈A and y∈B (iii) If A={1,2},B={3,4} then A×(B∩ϕ)=ϕ

Answer:

(i) True (ii) True (iii) False; A×(B∩ϕ)=ϕ is an empty set

Question:

If A×B={(a,x),(a,y),(b,x),(b,y)} Find A and B.

Answer:

A = {a,b} B = {x,y}

Question:

The cartesian product A×A has 9 elements among which are found (−1,0) and (0,1). Find the set A and the remaining elements of A×A.

Answer:

Set A = {-1, 0, 1}

The remaining elements of A×A are: (1, -1), (1, 0), (1, 1), (0, -1), (0, 0), (-1, 1), (-1, -1), (-1, 0)

Question:

If (3x​+1,y−32​)=(35​,31​) find the values of x and y

Answer:

1. Rearrange the equation to isolate the variable: 3x + 1 = 35

2. Subtract 1 from both sides: 3x = 34

3. Divide both sides by 3: x = 11

4. Substitute x = 11 into the original equation: 3(11) + 1 = 35

5. Simplify: 32 = 35

6. Subtract 32 from both sides: y = 31

Question:

Let A={1,2},B={1,2,3,4},C={5,6} and D={5,6,7,8} Verify that (i) A×(B∩C)=(A×B)∩(A×C) (ii) A×C is a subset of B×D

Answer:

(i) A×(B∩C) = { (1,1), (1,2), (2,1), (2,2) } (A×B)∩(A×C) = { (1,1), (1,2), (2,1), (2,2) }

A×(B∩C) = (A×B)∩(A×C)

(ii) A×C = { (1,5), (1,6), (2,5), (2,6) } B×D = { (1,5), (1,6), (1,7), (1,8), (2,5), (2,6), (2,7), (2,8), (3,5), (3,6), (3,7), (3,8), (4,5), (4,6), (4,7), (4,8) }

A×C is a subset of B×D

Question:

If the set A has 3 elements and the set B={3,4,5} then find the number of elements in (A×B)?

Answer:

Answer:

1. The set A has 3 elements, so A = {a1, a2, a3}.

2. The set B={3,4,5}, so B = {b1, b2, b3}.

3. The number of elements in (A×B) = 3 × 3 = 9.

Question:

If A = {-1, 1}, find A × A × A.

Answer:

A × A × A = {(-1) × (-1) × (-1), (-1) × (-1) × 1, (-1) × 1 × (-1), (-1) × 1 × 1, 1 × (-1) × (-1), 1 × (-1) × 1, 1 × 1 × (-1), 1 × 1 × 1}

= {-1, -1, -1, 1, -1, 1, -1, 1}

Question:

If A = {-1, 1}, find A × A × A.

Answer:

A × A × A = {(-1) × (-1) × (-1), (-1) × (-1) × 1, (-1) × 1 × (-1), (-1) × 1 × 1, 1 × (-1) × (-1), 1 × (-1) × 1, 1 × 1 × (-1), 1 × 1 × 1}

= {-1, -1, -1, 1, -1, 1, -1, 1}

Question:

Let A={1,2} and B={3,4}. Write A×B. How many subsets will A×B have?

Answer:

A x B = {(1,3), (1,4), (2,3), (2,4)}

A x B will have 4 subsets.

Question:

If G={7,8} and H={5,4,2}, find G×H and H×G.

Answer:

G×H = {(7,5), (7,4), (7,2), (8,5), (8,4), (8,2)}

H×G = {(5,7), (5,8), (4,7), (4,8), (2,7), (2,8)}

Question:

Let A and B be two sets such that n(A)=3 and n(B)=2. If (x,1),(y,2),(z,1) are in A×B find A and B where x,y and z are distinct elements

Answer:

A = {x,y,z} B = {1,2}