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# discretet1sol - DISCRETE MATH MIDTERM 1 Name You have 50...

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Unformatted text preview: DISCRETE MATH MIDTERM 1 Name: You have 50 minutes for this exam. If you have a question, raise your hand and remain seated. In order to receive full credit your answer must be complete , legible and correct . 1. What is the definition of the italicized word or phrase? (a) “ f is a function from A to B ”. f is a function from A to B if f ⊆ A × B and for all a ∈ A there is a unique b ∈ B such that ( a, b ) ∈ f . (b) “ P ( X ) is the power set of X ”. The power set of X is the set P ( X ) of all subsets of X . 2. Show that P ( A ∩ B ) = P ( A ) ∩ P ( B ). x ∈ P ( A ∩ B ) ⇔ x ⊆ A ∩ B ⇔ x ⊆ A and x ⊆ B ⇔ x ∈ P ( A ) and x ∈ P ( B ) ⇔ x ∈ P ( A ) ∩ P ( B ) 1 2 3. (a) What is the recursive definition of multiplication of natural numbers? (IV) m · 0 := 0 (RR) m · S ( n ) := m · n + m (b) Prove by induction that ℓ ( m + n ) = ℓm + ℓn for all ℓ, m, n ∈ N . (You may use any previously proved theorems that concern addition .) Let S n be the statement that “...
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