Relations and Functions

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Step 1: Cartesian product means all ordered pairs (a,b) where a ∈ A and b ∈ B. Step 2: From A = {1, 2}, first element can be 1 or 2. Step 3: From B = {x, y, z}, second element can be x, y, or z. Step

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Step 1: Use the formula n(A × B) = n(A) × n(B) Step 2: Given n(A) = 3 and n(B) = 4 Step 3: n(A × B) = 3 × 4 = 12 Answer: 12 ordered pairs Note: This works because each element in A pairs with each ele

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Step 1: Check definition of equal ordered pairs: (a,b) = (c,d) if and only if a = c and b = d Step 2: For (2,3) = (3,2), we need first elements equal and second elements equal Step 3: First elements:

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Step 1: Relation means y = 2x where x ∈ A and y ∈ B Step 2: Check each x in A: - For x = 1: y = 2(1) = 2, and 2 ∈ B ✓ → (1,2) - For x = 2: y = 2(2) = 4, and 4 ∈ B ✓ → (2,4) - For x = 3: y = 2(3) = 6,

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Step 1: Domain = set of all first elements in ordered pairs Step 2: From R = {(1,4), (2,5), (3,4), (4,6)} First elements are: 1, 2, 3, 4 Domain = {1, 2, 3, 4} Step 3: Range = set of all second element

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Step 1: For a relation to be a function, each element in domain must have exactly one image Step 2: Check domain elements: - Element 1 appears in: (1,2) and (1,4) - Element 1 has two images: 2 and 4 S

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Step 1: Check if it's a function - each domain element must have exactly one image Domain elements: 1, 2, 3 - 1 maps to 3 only ✓ - 2 maps to 5 only ✓ - 3 maps to 7 only ✓ Step 2: Yes, f is a functio

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Step 1: Given f(x) = 3x + 2, substitute values Step 2: Find f(4): f(4) = 3(4) + 2 = 12 + 2 = 14 Step 3: Find f(-1): f(-1) = 3(-1) + 2 = -3 + 2 = -1 Answer: f(4) = 14 and f(-1) = -1 Note: Always substi