Functions, Part 2

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Misc

Colloquium

“Structure and Symmetry: An Introduction to Lie Algebras and Representation Theory”

By Chad Magnum, Niagara University

Friday, April 7, 2:30 PM, Newton 203

Extra credit as usual for writing a paragraph or so about connections you make to the talk.

Questions?

Injections, Surjections, Bijections

Section 6.3

Injections

Is the “n^th odd number” function, i.e., f : ℕ → ℕ defined by f(n) = 2n - 1, an injection? How about g : ℝ → ℝ defined by g(x) = x³ - x?

Relevant ideas or questions from the reading

• Injection is one-to-one, i.e., every x produces a different y
• Formal definition: for all x₁ and x₂ in A, where f : A → B, if x₁ ≠ x₂ then f(x₁) ≠ f(x₂)

Solutions

• f is injective because each odd number seems to be produced by a distinct n value.
• g is not injective, because g(-1), g(0), and g(1) all produce the same value, 0.

Proofs

• That f is an injection: Show the contrapositive of the formal definition given above: if f(n) = f(m) then n = m. Suppose f(n) = f(m), i.e., 2n - 1 = 2m - 1. Then via algebra we see that 2n = 2m, and then n = m. This proves the contrapositive.
• That g is not an injection: Taken care of by the counterexample given earlier.

Comments

• “Injection” is defined as given in the reading summary above.
• You commonly prove that functions are injections via the contrapositive
  • ... Which is roughly equivalent to solving for x in an equation that defines f(x), and seeing if there is more than one solution.

Surjections

Which standard trignometric functions (sin, cos, tan, sec, csc, cot) are surjections from ℝ onto ℝ?

• Correction: domain, at least for tan, should be ℝminus the odd multiples of π/2; for cot domain should be ℝminus the even multiples of π/2.

Relevant ideas or questions from the reading

• Let f: A → B, then f is a surjection if and only if for every y in B there is an x in A such that f(x) = y.
• Not all functions are surjections, but any formula can define a function that is a surjection if the codomain is restricted to be exactly equal to the range.
• This definition implies that the codomain of a surjection equals the range.

Solutions

• Not surjections: sin, cos, sec, csc, because they don’t produce certain real numbers.
• Surjections: tan, cot.

Another example: Is f(n) = 2n -1 a surjection, given f : ℕ → ℕ?

• No, not all naturals are odd.
• Counterexample: 2 is not the image of any n under f.

What if we change the domain and ask about h(n) : ℚ → ℕ defined by h(n) = 2n - 1.

• h(n) is a surjection.
• Proof: let y be any natural number. Writing y as 2n -1 and solving for n yields n = (y+1)/2 which is rational, i.e. an element of ℚ that maps to y.

Comments

• The definition of surjection given on the first line of the reading summary is a good one.

Bijections

Are any of the functions above (i.e., f(n) = 2n - 1, g(x) = x³ - x, the standard trig functions) bijections?

Relevant ideas or questions from the reading

• A bijection is a function that is an injection and a surjection

Overall Comments

The most important ideas to take away from today are the definitions of injection, surjection, and bijection. You’ll see those terms a lot.

The second most important take-away is the methods for proving that functions are injections or surjections.

Next

Friday. “Class out of context” with Prof. Reuter

By definition, no reading, and there probably won’t be Canvas notes.

Monday. Composition of functions.

Read textbook section 6.4

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