1. If $\left ( \frac{x}{3}+1,y-\frac{2}{3} \right )=\left ( \frac{5}{3},\frac{1}{3} \right )$ ,find the values of x and y.
Since the ordered pairs are equal, corresponding elements will also be equal.
∴ $\left ( \frac{x}{3}+1 \right )=\frac{5}{3}$ and $\left ( y-\frac{2}{3} \right )=\frac{1}{3}$
⇒ $\frac{x}{3}=\frac{5}{3}-1$ and $y=\frac{1}{3}+\frac{2}{3}$
⇒ $\frac{x}{3}=\frac{2}{3}$ and $y=\frac{3}{3}$
⇒ x=2 and y=1
2. If the set A has 3 elements and the set B ={3, 4, 5}, then find the number of elements in (A×B).
It is given that n(A)=3
B={3,4,5} ∴ n(B)=3
Number of elements in (A×B) = n(A)*n(B) = 3*3 = 9
3. If G ={7, 8} and H ={5, 4, 2}, find G × H and H × G.
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)}
4. 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)}.
False
If P ={m, n} and Q ={ n, m}, then P⨯Q = {(m,n),(m,m),(n,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.
True
(iii) If A = {1, 2}, B = {3, 4}, then A × (B ∩ φ) = φ.
True
5. If A = {–1, 1}, find A × A × A.
A⨯A = {(-1,-1),(-1,1),(1,-1),(1,1)}
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)}
6. If A × B = {(a, x),(a , y), (b, x), (b, y)}. Find A and B.
We know that Cartesian product of A and B is,
A⨯B={(a,b): a∈A, b∈B} i.e. A is set of first elements and B is set of second elements
∴ A={a,b} and B={x,y}
7. 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)
B∩C = ϕ
A⨯(B∩C) = ϕ ...(i)
A⨯B = {(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(2,4)}
A⨯C = {(1,5),(1,6),(2,5),(2,6)}
(A⨯B)∩(A⨯C) = ϕ...(ii)
From (i) and (ii), A⨯(B∩C)=(A⨯B)∩(A⨯C)
(ii) A × C is a subset of B × D.
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)}
We can observe that elements of A⨯C are the elements of B⨯D.
∴ (A⨯C)⊂(B⨯D)
8. Let A ={1, 2} and B ={3, 4}. Write A × B. How many subsets will A × B have? List them.
A⨯B={(1,3),(1,4),(2,3),(2,4)}
n(A⨯B)= 4
we know that if n(A⨯B)=m, then n(P(A⨯B))=$\displaystyle \small 2^{m}$
∴ n(P(A⨯B))=$\displaystyle \small 2^{4}$=16 subsets.
{ϕ,{(1,3)}, {(1,4)},{(2,3)},{(2,4)},{(1,3),(1,4)},{(1,3),(2,3)},{(1,3),(2,4)},{(1,4),(2,3)},{(1,4),(2,4)},{(2,3),(2,4)},{(1,3),(1,4),(2,3)},{(1,3),(1,4),(2,4)},{(1,3),(2,3),(2,4)},{(1,4),(2,3),(2,4},{(1,3),(1,4),(2,3),(2,4)}}
9. 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.
It is given that, n(A)=3, n(B)=2, A⨯B={(x,1)(y,2),(z,1)}
We know that A is set of first elements and B is set of second elements.
∴ A={x,y,z} and B={1,2}
10. 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.
It is given that n(A⨯A)=9
we know that n(A⨯A)= n(A)*n(A)
∴ n(A)=3
we know that A⨯A={(a,a): a∈A} and ordered pair (-1,0) and (0,1) are two of the nine elements of A⨯A.
∴ A={-1,0,1}
remaining elements of A⨯A are (-1,-1),(-1,1),(0,-1),(0,0),(1,-1),(1,0),(1,1)
Since the ordered pairs are equal, corresponding elements will also be equal.
∴ $\left ( \frac{x}{3}+1 \right )=\frac{5}{3}$ and $\left ( y-\frac{2}{3} \right )=\frac{1}{3}$
⇒ $\frac{x}{3}=\frac{5}{3}-1$ and $y=\frac{1}{3}+\frac{2}{3}$
⇒ $\frac{x}{3}=\frac{2}{3}$ and $y=\frac{3}{3}$
⇒ x=2 and y=1
2. If the set A has 3 elements and the set B ={3, 4, 5}, then find the number of elements in (A×B).
It is given that n(A)=3
B={3,4,5} ∴ n(B)=3
Number of elements in (A×B) = n(A)*n(B) = 3*3 = 9
3. If G ={7, 8} and H ={5, 4, 2}, find G × H and H × G.
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)}
4. 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)}.
False
If P ={m, n} and Q ={ n, m}, then P⨯Q = {(m,n),(m,m),(n,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.
True
(iii) If A = {1, 2}, B = {3, 4}, then A × (B ∩ φ) = φ.
True
5. If A = {–1, 1}, find A × A × A.
A⨯A = {(-1,-1),(-1,1),(1,-1),(1,1)}
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)}
6. If A × B = {(a, x),(a , y), (b, x), (b, y)}. Find A and B.
We know that Cartesian product of A and B is,
A⨯B={(a,b): a∈A, b∈B} i.e. A is set of first elements and B is set of second elements
∴ A={a,b} and B={x,y}
7. 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)
B∩C = ϕ
A⨯(B∩C) = ϕ ...(i)
A⨯B = {(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(2,4)}
A⨯C = {(1,5),(1,6),(2,5),(2,6)}
(A⨯B)∩(A⨯C) = ϕ...(ii)
From (i) and (ii), A⨯(B∩C)=(A⨯B)∩(A⨯C)
(ii) A × C is a subset of B × D.
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)}
We can observe that elements of A⨯C are the elements of B⨯D.
∴ (A⨯C)⊂(B⨯D)
8. Let A ={1, 2} and B ={3, 4}. Write A × B. How many subsets will A × B have? List them.
A⨯B={(1,3),(1,4),(2,3),(2,4)}
n(A⨯B)= 4
we know that if n(A⨯B)=m, then n(P(A⨯B))=$\displaystyle \small 2^{m}$
∴ n(P(A⨯B))=$\displaystyle \small 2^{4}$=16 subsets.
{ϕ,{(1,3)}, {(1,4)},{(2,3)},{(2,4)},{(1,3),(1,4)},{(1,3),(2,3)},{(1,3),(2,4)},{(1,4),(2,3)},{(1,4),(2,4)},{(2,3),(2,4)},{(1,3),(1,4),(2,3)},{(1,3),(1,4),(2,4)},{(1,3),(2,3),(2,4)},{(1,4),(2,3),(2,4},{(1,3),(1,4),(2,3),(2,4)}}
9. 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.
It is given that, n(A)=3, n(B)=2, A⨯B={(x,1)(y,2),(z,1)}
We know that A is set of first elements and B is set of second elements.
∴ A={x,y,z} and B={1,2}
10. 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.
It is given that n(A⨯A)=9
we know that n(A⨯A)= n(A)*n(A)
∴ n(A)=3
we know that A⨯A={(a,a): a∈A} and ordered pair (-1,0) and (0,1) are two of the nine elements of A⨯A.
∴ A={-1,0,1}
remaining elements of A⨯A are (-1,-1),(-1,1),(0,-1),(0,0),(1,-1),(1,0),(1,1)
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