In each of 28-31: a. Rewrite the theorem in three different ways: as ∀ _______if_______then_______» ∀ _______________(without using the words if or then ) and as If_________, then________(without using an explicit universal quantifier). b. Fill in the blanks in the proof of the theorem. Theorm: The sum of any two odd integers is even. Proof: Suppose m and n are any [particulasr but arbitarity chosen] odd integers. [ We must show that m + n is even.] By (a) m + 2 r + 1 and n = 2 s + 1 for some integers r and s. Then m + n = ( 2 r + 1 ) + ( 2 s + 1 ) b y (b) =2 r + 2 s + 2 = 2 ( r + s + 1 ) b y algerta Hence m + n = 2 u where u is an integer, and so, by (d) m + n is even [ as was to be shown].
In each of 28-31: a. Rewrite the theorem in three different ways: as ∀ _______if_______then_______» ∀ _______________(without using the words if or then ) and as If_________, then________(without using an explicit universal quantifier). b. Fill in the blanks in the proof of the theorem. Theorm: The sum of any two odd integers is even. Proof: Suppose m and n are any [particulasr but arbitarity chosen] odd integers. [ We must show that m + n is even.] By (a) m + 2 r + 1 and n = 2 s + 1 for some integers r and s. Then m + n = ( 2 r + 1 ) + ( 2 s + 1 ) b y (b) =2 r + 2 s + 2 = 2 ( r + s + 1 ) b y algerta Hence m + n = 2 u where u is an integer, and so, by (d) m + n is even [ as was to be shown].
Solution Summary: The author explains how to rewrite the theorem "The sum of any two odd integers is even."
In each of 28-31: a. Rewrite the theorem in three different ways: as
∀
_______if_______then_______»
∀
_______________(without using the words if or then) and as If_________, then________(without using an explicit universal quantifier). b. Fill in the blanks in the proof of the theorem. Theorm: The sum of any two odd integers is even. Proof: Suppose m and n are any [particulasr but arbitarity chosen] odd integers. [We must show that
m
+
n
is even.] By (a)
m
+
2
r
+
1
and
n
=
2
s
+
1
for some integers r and s. Then
m
+
n
=
(
2
r
+
1
)
+
(
2
s
+
1
)
b
y
(b)
=2
r
+
2
s
+
2
=
2
(
r
+
s
+
1
)
b
y
algerta
Hence
m
+
n
=
2
u
where u is an integer, and so, by (d) m+n is even [as was to be shown].
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MFCS unit-1 || Part:1 || JNTU || Well formed formula || propositional calculus || truth tables; Author: Learn with Smily;https://www.youtube.com/watch?v=XV15Q4mCcHc;License: Standard YouTube License, CC-BY