Prove that If A1, A2, ... , An and B1, B2,...,Bn are sets such that Aj ⊆ Bj for j = 1, 2, 3, ... , n, then ∪j=1nAj ⊆ ∪j=1nBj .

This is proved using Proof by induction method. There are two steps in this method

Let P(n) represent the given statement  ∪ ⊆ ∪

1. Basis Step: This step proves the given statement for n = 1

2. Induction step: The step proves that if the given statement holds for any given case n = k  then it should also be true for n = k + 1.

If the above two steps are true this means that given statement P(n) holds true for all positive n and the mathematical induction P(n): ∪ ⊆ ∪ is true.

Step-by-step explanation:

Basis Step:

For n = 1

∪ = ∪ = A₁ ⊆ B₁ = ∪ = ∪

We show that

∪ = A₁ ⊆ B₁ = ∪  for n = 1

Hence P(1) is true

Induction Step:

Let P(k) be true which means that we assume that:

for all k with k≥1, P(k): ∪ ⊆ ∪ is true

This is our induction hypothesis and we have to prove that P(k + 1) is also true

This means if ∪ ⊆ ∪ holds for n = k  then this should also hold for n = k + 1.

In simple words if P(k): ∪ ⊆ ∪ is true then ∪ ⊆ ∪ is also true

∪ = ∪ ∪

           ⊆ ∪ ∪                 As ∪ ⊆ ∪

           ⊆ ∪ ∪                 As   ⊆

           =  ∪

The whole step:

∪ = ∪ ∪ ⊆ ∪ ∪ ⊆ ∪ ∪ =  ∪

shows that the P(k+1) also holds for ∪ ⊆ ∪

hence P(k+1) is true

So proof by induction method proves that P(n) is true. This means

P(n): ∪ ⊆ ∪ is true

the 3rd on is correct

step-by-step explanation:

60

step-by-step explanation:

every triangle measures a total of 120 degrees. you have to find angle c. we know that isosceles triangles angles are equal. 60 degrees plus 60 degrees gives you 120.