Name: SSN: Grade:MA334B EXAM 1 February 2000I pledge my honor that I have abided by the Stevens Honor System.1 (12pts)Let U = N = {0, 1, 2, . . .}. Are the following true or false? Explain.• ∃y ∀x [x + y = x]• ∀x ∃y [xy = x]• ∀x ∃!y [xy = yx]2 (16pts) True or false? If false, explain.1. Z ∩ R+= P2. Q ∪ Z = Z3. {2} ⊆ {1, 2}4. 2 ∈ {{1}, {2}}3 (20pts)Let A, B, C be sets. Prove by using the element method or disprove by a exhibiting a counterexample.1. (A ∪ B) x C ⊆ (A x C) ∩ (B x C)2. B ∩ (A ∪ C) ⊆ A ∪ (B ∩ C)4. (16pts)Let x and y be integers. Prove that if z = x + y is an odd integer, then either x is odd or y is odd.Hint: Use the contrapositive.5 (10pts)Use a truth table to check if the following is a tautology.[(p → q) ∧ (q → p)] → q6 (16pts)Let Σ = {a, b, c} be an alphabet. Let L1be the language consisting of all strings over Σ of length4 which begin with c and in which the letter b is always followed by the letter a. Let L2be thelanguage consisting of all palindromes (strings that read the same forward as backward) over Σ.1. List all elements of L1.2. Find L1∩ L2.7 (10pts)Let A = {1, 2, 3, 4, 5} and B = {a, b, c}.1. Partition AxB into 4 subsets.2. How large is the power set P