Foundations of Mathematics 30

Renting/leasing spreads smaller payments over time but you own nothing at the end; buying costs more up front but builds an asset.

Credit lets you buy now and pay later, but interest compounds on the unpaid balance until it is cleared.

A portfolio mixes investments with different risk and return; higher expected return usually means higher risk.

Total cost of ownership includes the price plus interest, insurance, maintenance, and how the item appreciates or depreciates.

A store card's '0% for 12 months' promotion often applies deferred interest retroactively if you miss the payoff date.

A conditional 'if p then q' has a converse (if q then p), an inverse (if not p then not q), and a contrapositive (if not q then not p).

A biconditional 'p if and only if q' is true only when p and q have the same truth value; it means both the conditional and its converse hold.

Strategies like guess-and-check, look-for-a-pattern, make-a-systematic-list, and work-backwards each fit different puzzle types.

A conditional and its contrapositive are always both true or both false; the converse and inverse need not match the original.

A truth table lists every combination of true/false to test when a compound statement holds.

Kinds of sets: empty set (∅), disjoint sets (no overlap), subset (⊂), and universal set (everything under consideration).

Complement A' is everything in the universal set not in A; intersection A ∩ B is what's in both; union A ∪ B is what's in either.

A Venn diagram shows how sets overlap; each region corresponds to a combination of 'and', 'or', and 'not'.

Overlap must be counted once: |A ∪ B| = |A| + |B| − |A ∩ B|.

Set notation names regions: A ∪ B (union/or), A ∩ B (intersection/and), A' (complement/not).

Probability is part-whole: favourable outcomes ÷ total outcomes, a value from 0 to 1.

Mutually exclusive events cannot happen together: P(A or B) = P(A) + P(B).

The fundamental counting principle: if one choice has m options and the next has n, together they have m × n.

Odds are part-part: favourable to unfavourable (e.g. 3:1 for) — not the same denominator as probability.

Non-mutually exclusive events can overlap: P(A or B) = P(A) + P(B) − P(A and B).

A polynomial function (degree ≤ 3) can rise and fall, so it models data that turns.

Exponential functions model constant-rate growth or decay (populations, investments, decay).

A sinusoidal function repeats: its period is how long one cycle takes and its amplitude is how far it swings.

The degree and the sign of the leading coefficient set the overall shape and end behaviour.

Logarithmic functions model quantities that grow fast then flatten, and underlie scales like Richter and pH.

A strong inquiry question is specific, answerable with data, and about a topic that matters to you or your community.

Data can be primary (you collect it) or secondary (already published); each needs its source checked.

A presentation (oral, written, or multimedia) organizes findings so an audience can follow the reasoning.

A rubric set up in advance names what a good research project looks like, so you can aim for it.

Assess accuracy, reliability, and relevance, and watch for bias and point of view.