And these of venn diagram

Mathematics

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Answers

5 parts
a
60
b
30
c
20
d
40
e
40

Step-by-step solution

10 steps

let's solve these Venn diagram questions.

Given:
n(U)=100n(U) = 100
n(A)=40n(A) = 40
n(B)=30n(B) = 30
n(A∩B)=10n(A \cap B) = 10

Part a· 2 steps

Find n(A∪B)n(A \cup B)

  1. Use the formula for the union of two sets.

    n(A∪B)=n(A)+n(B)−n(A∩B)n(A \cup B) = n(A) + n(B) - n(A \cap B)

  2. Substitute the given values.

    n(A∪B)=40+30−10n(A \cup B) = 40 + 30 - 10 n(A∪B)=70−10n(A \cup B) = 70 - 10 n(A∪B)=60n(A \cup B) = 60

    The number of elements in A∪BA \cup B is 60\boxed{60}.

Part b30

Find n(A only)n(A \text{ only})

  1. To find elements in A only, subtract the intersection from n(A)n(A).

    n(A only)=n(A)−n(A∩B)n(A \text{ only}) = n(A) - n(A \cap B)

  2. Substitute the given values.

    n(A only)=40−10n(A \text{ only}) = 40 - 10 n(A only)=30n(A \text{ only}) = 30

    The number of elements in A only is 30\boxed{30}.

Part c20

Find n(B only)n(B \text{ only})

  1. To find elements in B only, subtract the intersection from n(B)n(B).

    n(B only)=n(B)−n(A∩B)n(B \text{ only}) = n(B) - n(A \cap B)

  2. Substitute the given values.

    n(B only)=30−10n(B \text{ only}) = 30 - 10 n(B only)=20n(B \text{ only}) = 20

    The number of elements in B only is 20\boxed{20}.

Part d40

Find n(A∪B)′n(A \cup B)'

  1. The complement of A∪BA \cup B is found by subtracting n(A∪B)n(A \cup B) from the total number of elements in the universal set n(U)n(U).

    n(A∪B)′=n(U)−n(A∪B)n(A \cup B)' = n(U) - n(A \cup B)

  2. Substitute the values.

    We found n(A∪B)=60n(A \cup B) = 60 in part (a).

    n(A∪B)′=100−60n(A \cup B)' = 100 - 60 n(A∪B)′=40n(A \cup B)' = 40

    The number of elements in (A∪B)′(A \cup B)' is 40\boxed{40}.

Part e40

Find n(A′∩B′)n(A' \cap B')

  1. Apply De Morgan's Law, which states that A′∩B′=(A∪B)′A' \cap B' = (A \cup B)'.

    n(A′∩B′)=n((A∪B)′)n(A' \cap B') = n((A \cup B)')

  2. This is the same value calculated in part (d).

    n(A′∩B′)=40n(A' \cap B') = 40

    The number of elements in A′∩B′A' \cap B' is 40\boxed{40}.

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