Sets and Functions

Sets and Functions

1. If P = { 1, 2, 3 } and Q = { 4 , 5 } then what is the value of P × Q ?
1. { (4 , 5) , (1 , 5) , (2 , 4) , (2 , 5) , (3 , 4) , (3 , 5) }
2. { (1 , 4) , (4 , 5) , (2 , 4) , (2 , 5) , (3 , 4) , (3 , 5) }
3. { (1 , 4) , (1 , 5) , (2 , 4) , (3 , 5) , (3 , 4) , (3 , 5) }
4. { (1 , 4) , (1 , 5) , (2 , 4) , (2 , 5) , (2 , 4) , (3 , 5) }
5. { (1 , 4) , (1 , 5) , (2 , 4) , (2 , 5) , (3 , 4) , (3 , 5) }

1. Use the the Cartesian product X × Y formula.
The Cartesian product X × Y of sets X , Y is the set of all ordered pairs ( x , y ).
Two ordered pairs ( x1 , y1 ), ( x2 , y2 ) .

Correct Option: E

The Cartesian product X × Y of sets X , Y is the set of all ordered pairs ( x , y ).
Two ordered pairs ( x1 , y1 ), ( x2 , y2 ) .
Given that P = { 1, 2, 3 } and Q = { 4 , 5 } then,
P x Q = { 1, 2, 3 } x { 4 , 5 }
P x Q = { 1 } x { 4 , 5 } , { 2 } x { 4 , 5 } , { 3 } x { 4 , 5 }
P x Q = { 1 , 4 } , { 1 , 5 } , { 2 , 4 } , { 2 , 5 } , { 3 , 4 }, { 3 , 5 }
P x Q = { (1 , 4 ) , ( 1 , 5 ) , ( 2 , 4 ) , ( 2 , 5 ) , ( 3 , 4 ), ( 3 , 5) }

1. If A and B have n elements in common, how many elements do A × B and B × A have in common?
1. n
2. n3
3. n2
4. None of these

1. NA

Correct Option: C

NA

1. If U = {a, b, c, d, e, f}, A = {a, b, c}, B = {c, d, e, f}, C = {c, d, e}, find (A ∩ B) ∪ (A ∩ C).
1. {c}
2. {a}
3. {b}
4. {d}

1. A ∩ B = { a, b, c } ∩ { c, d, e, f }
A ∩ C = { a, b, c } ∩ { c, d, e }

Correct Option: A

A ∩ B = {a, b, c} ∩ {c, d, e, f}
A ∩ B = { c }
A ∩ C = { a, b, c } ∩ { c, d, e }
A ∩ C = { c }
∴ (A ∩ B) ∪ (A ∩ C) = { c }.

1. If U = {a, b, c, d, e, f}, A = {a, b, c}, find (U ∪ A)′.
1. U
2. A
3. Φ
4. None of these

1. U ∪ A = {a, b, c, d, e, f} ∪ {a, b, c}

Correct Option: C

U ∪ A = {a, b, c, d, e, f} ∪ {a, b, c} = {a, b, c, d, e, f} = U
(U ∪ A)′ = Φ.

1. Let A = {1, 3, 5} and B = {x: x is an odd natural number < 6}. Which of the following is false?
1. A ⊂ B
2. B ⊂ A
3. A = B
4. None of these

1. NA

NA