MAT246: Lecture 1B
Summer 2023
May 19th, 2023
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Announcements
⇤ HW1 has been released and is due Tuesday at 11:59 PM.
⇤ RA2 (due Tuesday 11:59) and week 2 material has been released.
⇤ The wiki and Wiki journal instructions have been released—see
Quercus!
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Warm-Up
Suppose that a right-angled triangle has side lengths of x, x+ 1, x+ 2
for some x 2 R. Show that x = 3.
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What’s wrong?
Proposition. For all integers a, b, c, if a | bc then a | b or a | c.
Proof. We assume that a, b, c 2 Z and that a | bc. So there exists a k 2 Z
such that bc = ka. Factor k as k = mn for m,n 2 Z, then:
bc = mna.
This means that b = ma or c = na, hence a | b or a | c. ⌅
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Proving Existential Statements: Direct Proof
How do you directly prove a statement like 9x . P (x)?
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Prove that if c, a, r 2 R such that c 6= 0 and r 6= a/c, then there exists
an x 2 R such that
ax+ 1
cx
= r.
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Proving Uniqueness
Prove that the x 2 R we found is unique.
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Proving Universal Statements: Direct Proof
How do you directly prove a statement like 8x . P (x)?
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Sometimes considering cases is easier
Prove that for all n 2 Z, 3n2 + n+ 14 is even.
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Proving Universal Statements: Proof by
Contradiction
How do you prove a statement like 8x . P (x) by contradiction?
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Proposition. Suppose x and y are odd integers. Then for all integers
z
x2 + y2 6= z2.
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Things we didn’t do
Proposition. For all n 2 Z, n is even if and only if 4 | n2.
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Determine whether each of the following statements is true or false. If a
statement is true, prove it. If a statement is false, provide a counterexample.
(a) The product of an odd integer and an even integer is odd.
(b) The product of an odd integer and an odd integer is odd.
(c) The product of an even integer and an even integer is even.
(d) The sum of an even integer and an odd integer is odd.