Question

3. Prove by contrapositive: Let n ∈ N. If n^3−5n−10>0,then n ≥ 3.

4. Prove: Letx∈Z. Then5x−11 is even if and only if x is odd.

4. Prove: Letx∈Z. Then 5x−11 is even if and only if x is odd.

Answer #1

3.a) Let n be an integer. Prove that if n is odd, then
(n^2) is also odd.
3.b) Let x and y be integers. Prove that if x is even and y is
divisible by 3, then the product xy is divisible by 6.
3.c) Let a and b be real numbers. Prove that if 0 < b < a,
then (a^2) − ab > 0.

Prove that every integer of the form 5n + 3 for n ∈ Z, n ≥ 1,
cannot be a perfect square

7. Prove by contradiction or contrapositive that for all
integers m and n, if m + n is even then m and n are both even or m
and n are both odd.

Prove the following: Let n∈Z. Then n2 is odd if and
only if n is odd.

Prove by induction that 3^n ≥ 5n+10 for all n ≥ 3.
I get past the base case but confused on the inductive step.

4. (a) Let n = (dkdk−1 . . . d0)b. Prove that (b + 1)|n if and
only if (b + 1)|d0 − d1 + d2 − · · · + (−1)kdk.
(b) A number is a palindrome if reversing the sequence of its
digits gives the same number. For example, 12321 and 456654 are
palindromes. Use part (a) with b = 10 to prove that every
palindrome with an even number of digits is divisible by 11.
(c) Which...

6. Consider the
statment. Let n be an integer. n is odd if and
only if 5n + 7 is even.
(a) Prove the forward implication of this statement.
(b) Prove the backwards implication of this statement.
7. Prove the following statement. Let a,b, and
c be integers. If a divides bc and
gcd(a,b) = 1, then a divides c.

Let x, y ∈Z. Prove that (x+1)y^2 is even if and only if x is odd
and y is even.

1. Let n be an integer. Prove that n2 + 4n is odd if and only if
n is odd? PROVE
2. Use a table to express the value of the Boolean function x(z
+ yz).

Write the contrapositive statements to each of the following.
Then prove each of them by proving their
respective contrapositives. In both statements assume x
and y are integers.
a. If the product xy is even, then at least one
of the two must be even.
b. If the product xy is odd, then both x and y
must be odd.
3. Write the converse the following statement.
Then prove or disprove that converse depending on whether it is
true or not. Assume x...

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