Prove each statement in 39-44. For every positive integer n , if A and B i , B 2 , B 3 . are any sets, then ∪ i = 1 n ( A × B i ) = A × ( ∪ i = 1 n B i )

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To determine To Prove: For all integers n ≥ 1 if A and B 1 , B 2 , B 3 , ...... are any sets, then ∪ i = 1 n ( A × B i ) = A × ( ∪ i = 1 n B i ) Explanation Given information: Let A and B 1 , B 2 , B 3 , .......... , B n ( n ≥ 1 ) are any sets. Concept used: ∪ : U n i o n of sets ∩ : I n t e r sec t i o n of sets ⊆ : subset of set ∅ : Null set Calculation: Let A and B 1 , B 2 , B 3 , .......... , B n ( n ≥ 1 ) are any sets. Prove that ∪ i = 1 n ( A × B i ) = A × ( ∪ i = 1 n B i ) First prove that ∪ i = 1 n ( A × B i ) ⊆ A × ( ∪ i = 1 n B i ) . Suppose there is an element ( x , y ) in ∪ i = 1 n ( A × B i ) , by definition of general union, ( x , y ) ∈ ( A × B i ) for some i = 1 , 2 , ..... , n Now by definition of Cartesian product, it follows that x ∈ A and y ∈ B i for some i = 1 , 2 , ..... , n . Since y ∈ B i for some i = 1 , 2 , ...... , n by definition of general union y ∈ ∪ i = 1 n ( B i ) . Since x ∈ A and y ∈ ∪ i = 1 n ( B i ) , by definition of Cartesian product, it follows that ( x , y ) ∈ A × ( ∪ i = 1 n B 1 ) . Thus, x ∈ ∪ i = 1 n ( A × B i ) ⇒ x ∈ A × ( ∪ i = 1 n B i ) Therefore, ∪ i = 1 n ( A × B i ) ⊆ A × ( ∪ i = 1 n B i )

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Answer 1

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BuyFind

Discrete Mathematics With Applicat...

5th Edition

EPP + 1 other

Publisher: Cengage Learning,

ISBN: 9781337694193

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Answer 3

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Answer 5

To determine

To Prove:

For all integers n≥1 if A and B1,B2,B3,...... are any sets, then

∪i=1n(A×Bi)=A×(∪i=1nBi)

Explanation of Solution

Given information:

Let A and B1,B2,B3,..........,Bn(n≥1) are any sets.

Concept used:

∪:Union of sets∩:Intersection of sets

⊆: subset of set

∅: Null set

Calculation:

Let A and B1,B2,B3,..........,Bn(n≥1) are any sets.

Prove that ∪i=1n(A×Bi)=A×(∪i=1nBi)

First prove that ∪i=1n(A×Bi)⊆A×(∪i=1nBi).

Suppose there is an element (x,y) in ∪i=1n(A×Bi), by definition of general union,

(x,y)∈(A×Bi) for some i=1,2,.....,n

Now by definition of Cartesian product, it follows that x∈A and y∈Bi for some i=1,2,.....,n.

Since y∈Bi for some i=1,2,......,n by definition of general union y∈∪i=1n(Bi).

Since x∈A and y∈∪i=1n(Bi), by definition of Cartesian product, it follows that

(x,y)∈A× (∪i=1nB1).

Thus, x∈∪i=1n(A×Bi)⇒x∈A×(∪i=1nBi)

Therefore, ∪i=1n(A×Bi)⊆A×(∪i=1nBi)

Prove each statement in 39-44. For every positive integer n , if A and B i , B 2 , B 3 . are any sets, then ∪ i = 1 n ( A × B i ) = A × ( ∪ i = 1 n B i )

Discrete Mathematics With Applicat...

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Discrete Mathematics With Applicat...