(a) If a | bc, show that a | b*gcd(a,c). (b) If a, b are coprime integers and c | at and c | bt, show that c | t. (c) If a, b, c are integers with a, c coprime, prove that gcd(ab, c) = gcd(b, c).
(c) Evaluate (A B)C and AT (BC) and show that they are equal. 4. Show that for any x the matrix [cos(2x) sin(2x) A= 4sin(2x) -cos(2x) Satisfies the relation A2 -I 5. Find the determinant of the following matrices:
SHOW WORK PLEASE
SHOW WORK PLEASE
The Boolean expression (A+B+C) (A+B'+C) has a dual which is: O a (A BH(A+B+C) O (ABC)+(A'B C) O c. (A+B+C) (A+B+C) odABC Oe (ABC) (A'BC)
5. (a) Show that Q(V2) C Q V2). (b) Find [Q( 12): Q(V2)]. (c) Show that r - V2 is irreducible in Q(V2)[].
Let A, B, C be three sets. Show that A ∪ B = A ∩ C ⇐⇒ B ⊆ A ⊆ C.
Show That
rn Show that | x Pn(C) Pn-l(x) dx =
3)Show that the truth function f(A, B, C) (A V B) C generates all truth functions. (Hint: Show that it generates two connectives that form adequate set.) an
3)Show that the truth function f(A, B, C) (A V B) C generates all truth functions. (Hint: Show that it generates two connectives that form adequate set.) an
Exercise 88 Show theatEC)()-C)-- 2" Exercise 89. Show that ) = 2"-1
Let a, b, c ∈ Z such that a|b and a|c. Show that a|kb + pc for any k, p ∈ Z.
Suppose a
c mod n and bd
mod n.
(a) show that a + b
c + d mod n
(b) show that a * b
c * d mod n.
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