Additional proofs: prove each statement below using proof by contradiction. 1. the sum of any rational number and any irrational number is irrational. s 2. for all integers m, if m is even, then 3m+7 is odd i integer, then, 2a +3b is even 3. if a is any odd integer, and b is any even 4. let a and b be sets from a universe u. if bc a, then ac b

See below.

Step-by-step explanation:

1.   Suppose that the sum is rational  then we can write:

a/b + i = c/d     where i is irrational and by definition a/b and c/d are rational.

Rearranging:

i = c/d - a/b

Now the sum on the right is rational  so 'irrational' = 'rational' which is a contradiction.

So  the original supposition is false and the sum must be irrational.

2. Proof of For all integers m if m is even then 3m + 7 is odd:

If m is even then 3m is even.

Suppose 3m + 7 is even, then:

3m + 7 = 2p  where p is an integer.

3m - 2p  = -7

But 3m and 2p are both even so their result is even  and -7 is odd.

Therefore the original supposition is false because it leads to a contradiction,  so 3m + 7 is odd.

mental cunning

explanation:

he is often referred to as a man of twists and turns

the answer to the expression is 83