Step-by-step explanation:
A direct proof is a method that takes an statement p, which we assume to be true, and use it to show directly that another statement q is true. So this method has the following steps:
Fact that we need to use:
Every odd integer can be written in the form 2m + 1 for some unique other integer m
Let p be the statement a and b be odd integers and q be the statement that the product of a and b is odd.
Proposition if a and b are odd, then the product of a and b is odd
Proof: Assume that a and b are odd integers, the by definition a = 2m + 1 and b = 2n + 1 for some integers m and n. we will now use this to show that the product of a and b is odd.
[tex]a\cdot b= (2m+1) \cdot (2n+1)\\a\cdot b = 2m\cdot 2n+2m+2n+1\\a\cdot b =4mn+2m+2n+1\\a\cdot b = 2(2mn+2m+2n) +1\\\:If \:k=2mn+2m+2n\\a\cdot b = 2k+1[/tex]
Hence we have shown that the product of a and b is odd since 2k + 1 is and odd integer. Therefore we have shown that p ⇒ q and so we have completed our proof.
Answer:
Step-by-step explanation:
The proof by the direct method that the product of two odd numbers is odd integer number, is the following:
Let [tex]z_1[/tex] and [tex]z_2[/tex] be two odd integers, then [tex]z_1 = 2a+1[/tex] and [tex]z_2 = 2b +1[/tex], for some integers a and b.
[tex]z_1z_2 = (2a + 1) (2b + 1)\\\\z_1z_2 = 4ab + 2a + 2b + 1\\\\z_1z_2 = 2 (2ab + a + b) +1\\\\z_1z_2 = 2n + 1[/tex]
where [tex]n = 2ab + a + b[/tex], which guarantees that [tex]n[/tex] is an integer number. In this way, [tex]z_1z_2[/tex] is an odd integer.
