Prove the following statement for non-zero integers a, b, c, If a divides b and b divides c, then a divides c. If an integer a divides an...
Take two integers, r and s. We can write r as a/b for integers a and b since a rational number can be written as a quotient of integers We can...
If a is an even integer and b is an odd integer then prove a − b is an odd integer Let a be our even integer Let b be our odd integer We can...
Let us take an integer x which is both even and odd. As an even integer, we write x in the form 2m for some integer m As an odd integer, we write...
Take two consecutive integers: n, n + 1 The difference of their cubes is: (n + 1)^3 - n^3 n^3 + 3n^2 + 3n + 1 - n^3 Cancel the n^3 3n^2 + 3n + 1...
Take two arbitrary integers, x and y We can express the odd integer x as 2a + 1 for some integer a We can express the odd integer y as 2b + 1 for...
Take an integer n. The next alternate consecutive integer is n + 2 Subtract the difference of the squares: (n + 2)^2 - n^2 n^2 + 4n + 4 - n^2...
Prove 0! = 1 Let n be a whole number, where n! represents: The product of n and all integers below it through 1. The factorial formula for n is...
Take an integer n. The next consecutive integer is n + 1 Subtract the difference of the squares: (n + 1)^2 - n^2 n^2 + 2n + 1 - n^2 n^2 terms...
Prove P(A’) = 1 - P(A) The sample space S contains an Event A and everything not A, called A' We know P(S) = 1 P(S) = P(A U A') P(A U A') = 1...
Use proof by contradiction. Assume sqrt(2) is rational. This means that sqrt(2) = p/q for some integers p and q, with q <>0. We assume p and q...
if a divides b, then a divides bc Suppose a divides b. Then there exists an integer q such that b = aq, so that bc = a(qc) and a divides bc....
If JK = PQ and PQ = ST, then JK=ST JK = PQ | Given Substitute ST for PQ since PQ = ST | Substitution JK = ST
if p=2x is even, then p^2 is also even p^2 = 2 * 2 * x^2 p^2 = 4x^2 This is true because: If x is even, then x^2 is even since two evens...
n^2+n = odd Factor n^2+n: n(n + 1) We have one of two scenarios: If n is odd, then n + 1 is even. The product of an odd and even number is an...
n^2-n = even Factor n^2-n: n(n - 1) We have one of two scenarios: If n is odd, then n - 1 is even. The product of an odd and even number is an...
if a and b are odd then a + b is even Let a and b be positive odd integers of the form: a = 2n + 1 b = 2m + 1 a + b = 2n + 1 + 2m + 1 a + b =...
if sc = hr and hr=ab then sc=ab sc = hr (given) Since hr = ab, we can substitute ab for hr by substitution: sc = ab
Given: 9 - 4x = -19 Prove: x = 7 Solve for x in the equation 9 - 4x = - 19 Step 1: Group constants: We need to group our constants 9 and -19....
if m is odd and n is odd, then mn is odd. m = 2k +a where a = 0 or 1 n = 2l + b where b = 0 or 1 mn = (2k + a)(2l + b) = 4kl + 2kb + 2al + ab...
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